Symbolic Computation Methods for Functional Analysis

Lecture WS 2005/2006

Markus.Rosenkranz@oeaw.ac.at

Symbolic Functional Analysis

As a significant body of research, the discipline of Symbolic Computation was created through the initiation of the Journal of Symbolic Computation by Prof. Bruno Buchberger in 1985 (he was chief editor until 1995). In the editorial of the first journal issue [4], Symbolic Computation is characterized as the "algorithmic solution of problems dealing with symbolic objects". It is described as as a bridge between two important main streams of algorithmic mathematics: computation and deduction. Accordingly, there are two major subcommunities in Symbolic Computation, which we could call Computational Algebra (or "computer algebra", but in a sense that is broader than what we discuss in the sequel under this name) and Computational Logic (also known as "automated theorem proving"). The integration of these two streams is still a challenge today, witnessed by recent EU initiatives like the Calculemus Project [2], whose express purpose is to combine the power of computer algebra systems and automated deducation systems. On a somewhat more precise basis, a unified definition of Symbolic Computation can be given in the following way [3], where "isomorphic" should be understood in a rather broad sense (meaning that certain properties considered relevant are reflected faithfully under an otherwise arbitrary correspondence):

Let us give two key examples, one for Computational Algebra and one for Computational Logic. For the latter, the main problem of automated theorem proving itself is a case in point: Given a goal formula G and a knowledge base K, test whether G follows from K. Of course, one has to first make precise what we mean by "G follows from K". What is nowadays a triviality was a major breakthrough in the realm of Mathematical Logic, when in 1933 Alfred Tarski gave a definition that is now universally accepted [5]: A formula G follows from a set of formulae K iff G is true in all models that make K true. For a precise formulation of "truth", "model" and related concepts, we refer to [6]. The important point for us here is the following: According to Tarski's definition (also known as the semantic entailment characterization), it is impossible to check whether G follows from K by an algorithm—we would have to search through the infinite list of models that makes K true (each typically being infinite itself) and then check whether G is also true in it. Fortunately, there is an algorithmic alternative to Tarski's approach, known as the syntactic entailment characterization: One sets up a proof of G from the axioms in K. Although the art of mathematical proving was known for more than two thousand years, it was only in 1930 that the Austrian logician Kurt Gödel proved that these two concepts are equivalent (using a kind of informal semantic characterization that was soon accomodated to Tarski's definition); this is the famous Completeness Theorem of mathematical logic. Unlike its model–theoretic counterpart, the proof search can be automated, and there are nowadays already numerous computer systems dedicated to this purpose. Let us mention one of these, which was developed in our own institute (RISC), under the supervision of Prof. Bruno Buchberger, named Theorema. This system is especially target at natural–style proving with support in various other areas relevant for the working mathematician (solving, computing, publishing, experimenting). A final word of warning about the theoretical and practical limitations of proving. We have said that there are nowadays several computer programs for proving theorems, but this does not mean that the main problem of automated theorem proving is completely solved:

Now let us turn to Computational Algebra, a branch of Symbolic Computation that could be characterized by its restriction (cum grano salis) to algebraic domains. What is an algebraic domain? Well, first of all: what is a domain, anyway? Let us first list a few domains (also called "structures") to get a feeling for this concept: We can think of the real numbers, any vector space, a topological space, a Hilbert space, a group, a ring, a partially ordered set (also known as "poset"), and a lattice. Each of these is specified by its so–called signature, i.e. the type of its operations, and its axioms. For example, a group is a domain G with a kind of multiplication ·:G×G -> 
        G, a neutral element 1 : G, and an inversion (…^(-1)) : G -> G, so its signature would by the list \(TraditionalForm\`⟨\(·\)\(\(,\)\(1\)\(,\)\(…\^\(-1\)\)\)⟩\); the axioms are of course given by associativity x (y z) = (x y) z, unitality 1  x = x, and invertibility x (x^(-1)) = 1. Also note that the system of real numbers can be conceived as a domain in various ways, depending on which operations and properties we want to include: an additive group, a multiplicative group, a ring, a field, an ordered field, a topological space, etc. Now which of these domains are algebraic? Looking at a typical treatment of algebra, we observe that it deals primarily with those domains whose signatures have only functions but no predicates and whose axioms are all identities. Hence a group is typical algebraic domain, having binary operation ·, a unary operation (…^(-1)) and a constant 1 (which can be seen as a "nullary operation"), together with three identities as axioms. In the list given above, all domains are alebraic except: the real numbers (if we include e.g. its order relation or topology), topological spaces, Hilbert spaces, posets. For example, a topological space needs a predicate of the form "is an open set", a Hilbert space has non–equational conditions on the inner product (in particular, its completeness), and a poset is based on a binary relation  ≤ . One should not be too dogmatic with this distinction, though. For example, fields are usually considered to be domains of algebra even though the axiom x ≠ 0 => x (x^(-1)) = 1 is a conditional identitiy rather than a plain identity.
Among the most important domains of Computational Algebra are the rings, hence it is no surprise that ring theory makes up a large part of this area. One of the central notions in this theory is that of an ideal (for the sake of simplicity we assume commutative rings here): For a ring R, an ideal is a subset I ⊆ R such that I is a subgroup of the additive group of R and closed under arbitrary multiples, i.e. we have a x ∈ I for all a ∈ I and all x ∈ R. This notion is especially interesting if the ring under consideration is the ring of polynomials over some field K, say in three indeterminates x and y; this is usually written as R = K[x, y, z]. In such a ring, an ideal is nothing else than a collection of polynomial equations. Usually these are given by a finite list of basis equations (Subscript[g, 1]) = 0 ∧ … ∧ (Subscript[g, n]) = 0, and the ideal I is then formed by taking any R–linear combination of these basis equations, so the ideal is given as I = {(Subscript[p, 1])(Subscript[g, 1]) + … + 
            (Subscript[p, n])(Subscript[g, n])}; one says that ⟨(Subscript[g, 1]), …, 
        (Subscript[g, n])⟩ generates I. By the Hilbert Basis Theorem, any ideal has such a finite set of generators {(Subscript[g, 1]), …, (Subscript[g, n])}. Note that one can always write a polynomial equation in the form … = 0, so that one usually identifies the equation with the polynomial on its left–hand side. The (infinite!) ideal is, so to say, the most neutral way of characterizing such a system of equations: By forming various linear combinations, one can always come up with many different equational systems that are all equivalent—they all generate one and the same ideal, which is simply the collection of all possible equations.
Now ideals in polynomial rings are of special importance because they are an algebraic specification of various interesting geometric sets, called varieties, most notably curves and surfaces if one thinks of three–dimensional space. The variety given by an ideal I is simply the set of all those points \(TraditionalForm\`⟨ξ, η, ζ⟩\) where p(ξ, η, ζ) = 0 for all polynomials p ∈ I, so it contains just the zeroes of the system of equations encoded in I. Obviously it is enough to require g(ξ, η, ζ) = 0 for the generators g of I. As an example, the variety of ⟨(x^2) + (y^2) + (z^2) - 
          1⟩ is the unit sphere in (ℝ^3).
There is one problem that plays a crucial role in this area because many geometric questions reduce to it, the so–called main problem of ideal theory; see [8]: Let I be an ideal in a polynomial ring R = K[(Subscript[x, 1]), …, (Subscript[x, s])] given by an ideal basis ⟨(Subscript[b, 1]), …, 
        (Subscript[b, n])⟩. Now given an arbitrary polynomial p ∈ K[(Subscript[x, 1]), …, (Subscript[x, s])], test whether or not p ∈ I. In other words, we want to know whether or not the polynomial equation p = 0 follows from the polynomial equation system (Subscript[b, 1]) = 0 ∧ … ∧ (Subscript[b, n]) = 0. In a straight–forward approach, one would have to search through all the infinitely many R–linear combinations of (Subscript[b, 1]), …, (Subscript[b, n]), which is of course not algorithmically feasible. In order to obtain an algorithmic criterion, Bruno Buchberger has introduced the notion of a Gröbner basis for an ideal I in his doctoral thesis [9]; see [10] for a detailed survey of main ideas and applications and [14, 15] for an in–depth treatment of the theory. The notion of Gröbner basis for polynomial rings like K[(Subscript[x, 1]), …, (Subscript[x, s])] as well as for more general types of polynomials will be treated to some extent later in these lecture notes. For the moment, let us just say that a basis ⟨(Subscript[g, 1]), …, 
        (Subscript[g, m])⟩ for the ideal I is called a Gröbner basis if it fulfills some special structural properties. It can be used for an algorithmic solution of the ideal memebership problem as stated above because of the following two crucial constructions: Given any basis ⟨(Subscript[b, 1]), …, 
        (Subscript[b, n])⟩ of the ideal I, there is an algorithm for computing a Gröbner basis ⟨(Subscript[g, 1]), …, 
        (Subscript[g, m])⟩ of that same ideal. And given such a Gröbner basis, there is a simple algorithm that "reduces" the given input polynomial p to its normal form \(TraditionalForm\`p\&_\) with respect to the computed Gröbner basis. Then p ∈ I iff \(TraditionalForm\`p\&_ = 0\). One of the main applications of the ideal membership problems is to decide whether a "big" variety contains a "small" variety (e.g. whether some surface contains a certain curve): Essentially, this can be decided by checking whether the polynomials describing the big variety are members of the ideal describing the small variety. Hence one can translate certain important questions about infinite and, per se, non–algorithmic domains (like varieties) into algorithmically solvable tests (like polynomial reduction) on suitable "isomorphic" domains (like Gröbner bases).
The two problems stated above—automated theorem proving and ideal membership testing—give us a flavor of what symbolic computation is all about: If we "encode" a problem in the "right" way, it becomes accessible to symbolic solution methods, i.e. we can compute solution to the problem by manipulation of "symbols" (e.g. formulae and proofs, or polynomials and Gröbner bases). In this lecture, our main focus will be computational algebra, so let us analyze this field a bit closer. Of course, there are many ways of partitioning this field (which has grown quite large in the last decades), but for our present purposes we suggest the following three–level hierarchy based on the type of its fundamental objects:

See the picture below (sorry for the low quality due the conversion of PS to JPG!) for a schematic overview of this hierarchy, including some typical datastructures used there (to be explained in the lecture). In this picture, we have used the terms "symbolic algebra" as a synonym for "computer algebra" and "symbolic analysis" as a synonym for "computer analysis".

Naturally, this classification is rather crude, and we must also emphasize that the three "floors" introduced by our type scheme are by no means a scale of difficulty. This becomes clear by considering their classical (non-symbolic) counterparts: Roughly speaking, they are number theory, real analysis, functional analysis; now nobody would say that number theory is the "trivial basis" for analysis…
Let us now summarize our discussion by defining the scope of symbolic functional analysis as follows:

Let us comment briefly on the last sentence of this characterization. Obviously, one of the most important "computational" questions that one can ask about operators is whether and how they can be inverted. So given an operator A, we want to find another operator \(TraditionalForm\`A\&~\) such that \(TraditionalForm\`A\ A\&~ = 1\) and \(TraditionalForm\`A\&~\ A = 1\), where 1 denotes the identity operator. In the sequel, we will have occasion to view more general concepts of inversion. Beyond inversion in all its aspects, we can consider the problem of solving a system of operator equations; the two equations \(TraditionalForm\`A\ A\&~ = \(A\&~\ A = 1\)\) are just one example. So this would constitute a typical question of symbolic functional analysis: solving for operators by a constructive method.
As mentioned before, symbolic functional analysis (in this sense) is a very new, emerging field. Therefore there are no textbooks (yet) written on this field. However, some of the older books on "operator calculus" or "operator theory" (at least its constructive fragment) could be seen as pioneering steps in this direction. See for example [12] and [13], which also serve as a basis for a large part of this lecture (other sources are indicated where it seems significant).

Generalized Inversion of Linear Operators

Basic Set Theory and Linear Algebra

Let f ∶ X -> Y be any function with kernel K and image I. Then there is a canonical decomposition \(TraditionalForm\`f = \(ι\_\(I/
                Y\)\) \(f\&~\) π\_\(X/K\)\), where (Subscript[π, X/K]) ∶ X -> X/K is the canonical projection onto the coimage (the quotient "domain modulo kernel"), \(TraditionalForm\`f\&~ ∶ M/K -> I\) is the canonical isomorphism (known from the Isomorphism Theorem of algebra), and (Subscript[ι, I/Y]) ∶ I -> Y is the canonical insertion of the image. Note that (Subscript[π, X/K]) is surjective, \(TraditionalForm\`f\&~\) bijective, and (Subscript[ι, I/Y]) injective. In the sequel, it will also prove useful two introduce the two abbreviations (f^(↑I))=\(f\&~π\_\(X/K\)\) and \(TraditionalForm\`f\_\(\(↓\)\(K\)\) = ι\_\(I/Y\)f\&~\) for what we will call the image restriction of f and the coimage embedding of f, respectively. Sometimes we will also write briefly (f^↑) for (f^(↑I)) and (Subscript[f, ↓]) for (Subscript[f, ↓K]). Observe again that (f^(↑I)) is surjective and (Subscript[f, ↓K]) injective. Furthermore, we have the decompositions \(TraditionalForm\`f = ι\_\(I/Y\)f\^\(\(↑\)\(I\)\)\) and f = 
        (Subscript[f, ↓K])(Subscript[π, X/K]). Here is a diagrammatic representation of these decompositions (two commuting triangles):

f∶  ({{ , ↗, (Overscript[⟶, (f^(↑I))]), ↘,  ,  ,  }, {X, (Overscript[⟶, (Subscript[π, X/K])]), X/
              K, \(⟶\&\(f\&~\)\), I, (Overscript[⟶, (Subscript[ι, I/Y])]), Y}, { ,  ,  , ↘, (Underscript[⟶, (Subscript[f, ↓K])]), ↗,  }})

Incidentally, we can also use the notation for projections and insertions in a more general setting: For any projectable equivalence K', we may write (Subscript[π, X/K']) for the corresponding projection, and for any insertable subobject I', we may write (Subscript[ι, I'/Y]) for the corresponding insertion. For plain sets, the projectables are equivalences on X coarser than K and the insertables any subsets of Y containing I, whereas for groups, the projectables are the normal subgroups between K and X and the insertables any subgroups between I and Y, etc. This generalization extends also to restrictions and embeddings since they are defined in terms of such projections and insertions. We might call them codomain restrictions (narrowing f into some I') and quotient embeddings (collapsing f on some K'), respectively.
Let us emphasize that the usual notion of restriction for a function f ∶ X -> Y does not coincide with either restrictions or embeddings as introduced above. Given any subset—or subobject in a more general setting—of X, say X', the restriction \(TraditionalForm\`f\( | \_\(X'\)\)\(\(∶\)\(X' -> Y\)\)\) is given by the composite f(Subscript[ι, X'/X]). Note that \(TraditionalForm\`\(\(f\)\(\)\( | \_\(X'\)\)\)\) is restricted on the domain side (unlike the codomain restrictions) with X' being an immediate subset—or subobject—of the domain (unlike the quotient embeddings).
If rs = (Subscript[1, Y]) for two functions r ∶ X -> Y and s ∶ Y -> X, we say that r is a left inverse of or a retraction for s, and s is a right inverse or or a section for r. In a more symmetric manner, we will also say that the pair ⟨r, s⟩ is a cross section (with r being the "cross" and s being the "section"), more precisely a cross section of the identity on Y. (Note that in the literature, the term cross section is synonymous with section.) Obviously all retractions are surjective and all sections injective.
A function f ∶ X -> X is called idempotent if (f^2) = f; such functions deserve special attention. First of all, we may observe that any cross–section ⟨r, s⟩ with r ∶ X -> Y and s ∶ Y -> X gives rise to an idempotent function sr ∶ X -> X since we have

\(\((sr)\)\^2 = \(\(s\((rs)\)\)r = \(s1\_Yr = 
          s\(\(r\)\(\ \)\(.\)\)\)\)\)

\(\(\(f\^2 = \(ιf\&~πιf\&~π = \(ιf\&~π = 
          f\)\)\)\(,\)\)\)

These two identities are characteristic for the more special case of projectors, i.e. those idempotent functions that are at the same time linear endomorphisms on a module. If P ∶ M -> M is such a projector with kernel K and image I, we say that T projects along K onto I. In this setting, the above two identities have the following meaning: The projector does not change any elements already in its (co)image. So if P is as above, we have Px = x for all x ∈ I.
A submodule S of a module M is said to have an algebraic complement S' if S + S' = M and S ⋂ S' = O. Here S + S' denotes the set sum {s + s' | 
          s ∈ S ∧ s' ∈ S'} and O the zero module {0}. In such a case, we also write \(TraditionalForm\`M = S\( + \& . \)S'\) and we say that M is the (inner) direct sum of S and S'; it should be clear, however, that this is only a short–hand for a ternary predicate without involving any binary function \(TraditionalForm\`+\& . \). On the other hand, we may form the direct product S×S' of the modules S and S' by endowing their cartesian product with its natural module structure. Now if S and S' are algebraic complements (briefly called complements), there is a canonical isomorphism S×S' ≃ M. Sometimes the direct product of S and S' is also denoted by S⊕S' and accordingly called the (outer) direct sum. This apparently confusing situation is due to the fact that M is what categorists call a biproduct: Up to isomorphism, it is simultaneously a "product object", namely the product module S×S', and at the same time a "coproduct object", namely the sum module S + S'. Hence it comes natural to write S⊕S' ≃ S + S', and we will freely use both the direct product × and the direct sum ⊕ according to what fits better within a certain situation.
In the special case of vector spaces (modules over a field rather than just a ring), algebraic complements (there also known as complementary subspaces) always exist: Given a subspace U of vector space V, we can always complete a basis {(Subscript[b, 1]), …, (Subscript[b, k])} of S to a basis {(Subscript[b, 1]), …, (Subscript[b, k]), (Subscript[b, k + 1]), …, 
        (Subscript[b, n])} of all of V; then the added basis vectors from {(Subscript[b, k + 1]), …, (Subscript[b, n])} span the complement S'. In the more general context of modules, however, a complement need not always exist. Here is an example from abelian groups (seen as ℤ–modules): Obviously (Subscript[ℤ, 4]) = {0, 1, 2, 3} contains (Subscript[ℤ, 2])' = {0, 2}, an isomorphic copy of (Subscript[ℤ, 2]), as a subgroup and hence submodule. But (Subscript[ℤ, 2])' does not have a complement since {1, 3} cannot even be equipped with a group structure. Despite the generic failure, many practically relevent modules are nevertheless direct summands, i.e. they do have complements: most importantly, the so–called injective modules and free submodules of injective modules. This includes the important case of submodules in a free module, thus generalizing the situation of vector spaces.
Every direct summand S of a module M comes with its induced projector. If S + S' = M, every element x ∈ M has a unique decomposition x = s + s' with s ∈ S and s' ∈ S'. Then x ↦ s gives a projector P ∶ M -> M along S' onto S, whereas x ↦ s' is the projector P' ∶ M -> M along S onto S'; obviously one has P + P' = (Subscript[1, M]). As noted above, there is an isomorphism (Subscript[i, M]) ∶ M -> S×S'; which yields upon "division" two other isomorphisms (Subscript[i, S]) ∶ M/S' -> S and (Subscript[i, S']) ∶ M/S -> S'. In terms of these, the projectors are also given in their canonical decomposition as P = (Subscript[π, M/S]) (Subscript[i, S]) (Subscript[ι, S/M]) and P' = (Subscript[π, M/S']) 
          (Subscript[i, S']) (Subscript[ι, S'/M]). Conversely, every projector P ∶ M -> M gives rise to a direct sum \(TraditionalForm\`M = S\( + \& . \)S'\) with S = Im P and S' = Ker P. Hence one can say that projectors and complements amount to the same thing.
An outer direct sum M = (Subscript[M, 1])⊕(Subscript[M, 2]) yields an inner direct sum if one embeds the direct summands into the sum by the injections (Subscript[ι, 1]) ∶ (Subscript[M, 1]) -> M and (Subscript[ι, 2]) ∶ (Subscript[M, 2]) -> M given through x ↦ ⟨x, 
          0⟩ and x ↦ ⟨0, 
          x⟩, respectively. Writing (Subscript[M, 1])' and (Subscript[M, 2])' for the images of (Subscript[M, 1]) and (Subscript[M, 2]) under these injections, we obtain the inner direct sum \(TraditionalForm\`M = M\_1'\( + \& . \)M\_2'\). On the other hand, we can always go back from inner to outer summands by the projections (Subscript[π, 1]) ∶ M -> (Subscript[M, 1])' and (Subscript[π, 2]) ∶ M -> (Subscript[M, 2])' given through ⟨(Subscript[x, 1]), 
          (Subscript[x, 2])⟩ ↦ (Subscript[x, 1]) and ⟨(Subscript[x, 1]), 
          (Subscript[x, 2])⟩ ↦ (Subscript[x, 2]), respectively.  More precisely, this means we have \(TraditionalForm\`π\_1ι\_1 = 1\_\(M\_1\)\) and (Subscript[π, 2])(Subscript[ι, 2]) = 
        (Subscript[1, (Subscript[M, 2])]), so (Subscript[M, 1]) and (Subscript[M, 2]) are the images of (Subscript[M, 1])' and (Subscript[M, 2])' under the projections. As with all linear cross sections, they give rise to projectors (Subscript[P, 1]) = (Subscript[ι, 1])(Subscript[π, 1]) ∶ 
          M -> M and (Subscript[P, 2]) = (Subscript[ι, 2])(Subscript[π, 2]) ∶ 
          M -> M; obviously one has (Subscript[P, 1]) + (Subscript[P, 2]) = (Subscript[1, M]), with (Subscript[P, 1]) projecting along (Subscript[M, 2])' onto (Subscript[M, 1])' and (Subscript[P, 2]) projects along (Subscript[M, 1])' onto (Subscript[M, 2])'. Hence \(TraditionalForm\`P\_1P\_2 = \(P\_2P\_1 = 0\)\) or \(TraditionalForm\`π\_1ι\_2 = \(π\_2ι\_1 = 0\)\) upon cancelling injections left and the projections right. One can summarize and generalize all these relations by \(TraditionalForm\`π\_iι\_k = δ\_\(ik\)\) and ∑(Subscript[ι, i])(Subscript[π, i]) = 
        (Subscript[1, M]), where M = 
        (Subscript[M, 1])⊕…⊕(Subscript[M, n]) is multiple direct sum and the Kronecker delta is given by (Subscript[δ, ii]) = (Subscript[1, (Subscript[M, i])]) and (Subscript[δ, ik]) = 
        0 ∶ (Subscript[M, k]) -> (Subscript[M, i]) for i ≠ k.
Direct sums may also be formed for infinitely many modules, but then the isomorphism to direct products breaks down: Given a sequence of modules \(TraditionalForm\`⟨M\_i⟩\_\(i ∈ I\)\), their direct sum, usually denoted by (Subscript[⊕, i ∈ I])(Subscript[M, i]), is defined to be the submodule of all those elements ⟨(Subscript[m, 0]), 
        (Subscript[m, 1]), …⟩ of the direct product (∏_(i ∈ I))(Subscript[M, i]) that have finite support {i ∈ I | 
          (Subscript[m, i]) ≠ 0} and one may show that, in general, (Subscript[⊕, i ∈ I])(Subscript[M, i]) is a proper submodule of (∏_(i ∈ I))(Subscript[M, i]). If all modules in the sequence \(TraditionalForm\`⟨M\_i⟩\_\(i ∈ I\)\) are the same module I, their direct product is also written as (M^I) and their direct sum as (M^((I))). If furthermore I = ℕ, we write (M^∞) for (M^I) and (M^((∞))) for (M^((I))).

Codomain and Domain Refinements

A classical solution to this problem is provided by the well–known Isomomorphism Theorem (which is valid in the more general context of groups and even plain sets): If T ∶ M -> N is any epimorphism between modules, there is an isomorphism \(TraditionalForm\`T\&_ ∶ M/K -> N\), where K = Ker T. Here the idea is that we coarsen the domain such that it retains only the information that is not lost by T. But in operator theory, it turns out to be more practical, in some situations, if we proceed dually: Instead of coarsening the domain, we may as well refine the codomain so that it keeps all the information that is lost when M is carried into N by T, thus giving rise to a new operator \(TraditionalForm\`T\&_ ∶ M -> N×K\), that we will call the codomain refinement of T. In the classical case of ∂ on (C^∞)[0, 1], this amounts to keeping the constants in the "extra slot" K, so that \(TraditionalForm\`∂\&_\(\(∶\)\(\(C\^∞\)[0, 
              1] -> \(C\^∞\)[0, 
                1]×ℝ\)\)\) would map (x^2) + 1 to ⟨2  x, 1⟩ and (x^2) - 3 to ⟨2  
          x, -3⟩. Note that the isomorphism M ≃ N×K can be formally obtained from the homomorphism theorem M/K ≃ N by "multiplying" K from left to right.
Unlike in the Isomorphism Theorem, however, the construction of such an isomorphism depends on the choice of a particular complement L of the kernel K. As mentioned in the preparatory section, the kernel K may or may not have a complement in general, but for many important cases—among them vector spaces like (C^∞)[0, 1] and other function spaces—they do have a complement.

The submodule N×0 of N×K is isomorphic to N; if one identifies it with N itself, the definition of \(TraditionalForm\`T\&_\) in the proof may be written into the form \(TraditionalForm\`T\&_x = 
        Tx + T\&_x\_0\); this is the so–called fundamental formula of operator theory, cf. page 50 of [12]. It will play a crucial role in the development of the Mikusi(Overscript[n, '])ski calculus, where one takes ∂ for the operator T.
There is another situation, dual to the previous one, where inversion fails: if the given operator T is not surjective. In order to remedy this flaw, one has to add the missing preimages into the domain (just as we added the missing images of the kernel into the codomain in the previous construction); the isomorphic operator \(TraditionalForm\`T\&_\) thus obtained we will call the domain refinement of T. Of course, this means that we need a complement of the image space. We report the corresponding result here for reasons of symmetry, but we will not need it in the sequel—the case of restoring injectivity is much more important.

Of course, one can also deal with the situation of a homomorphism T ∶ M -> N that is neither injective nor surjective, and the obvious thing to do is to construct the (two–sided) refinement as \(TraditionalForm\`T\&_ ∶ M×J -> N×K\), assuming we have complements both for the kernel and the image. We will postpone the treatment of this case until we have the powerful concept of the Moore–Penrose inverse at hand; this will make the proof substantially simpler and more natural.
Instead of removing both obstracles to bijectivity by a simultaneous refinement, one may also use only one of either refinements on a suitably "mutilated" version of the homomorphism T ∶ M -> N. Again choosing a complement L for K = Ker T or a complement J for I = Im T, we apply Proposition  to (T^(↑I)) ∶ M -> I to obtain an isomorphism

M ≃ Im T×Ker T ,

N ≃ Im T×Coker T .

M×Coker T ≃ 
      N×Ker T ,

Let us also point out that the refinements go hand in hand with the corresponding complements in the following sense:

Sections and Retractions

In order to derive a necessary and sufficient condition for the existence of a kernel complement, we reformulate Proposition  in a more explicit manner, exposing the inverse of the isomorphism \(TraditionalForm\`T\&_\) as a section S ∶ N -> M. Of course, for obtaining the full action of the inverse, one has to "drag along" the kernel elements in the sense that \(TraditionalForm\`T\&_\^\(-1\) = S + 1\_K\).

We can also prove a dual result for monomorphisms, thus providing a necessary and sufficient condition for the existence of an image complement (which is slightly less important than for the kernel but still interesting for topological generalizations). Again, this gives us a representation for the inverse of the corresponding refinement \(TraditionalForm\`T\&_ ∶ M×J -> N\), namely \(TraditionalForm\`T\&_\^\(-1\) = R + 1\_J\). The proposition below contains Lemma 3.17 of page 32 in [19] as a special case.

From Corollary  we can see that there is a bijective correspondence between complements (of kernel / image) and semi–inverses (sections / retractions). In detail:

({{T∶, ({{ , ↗, (Overscript[⟶, (T^↑)]), ↘,  ,  ,  }, {M, (Overscript[⟶, (Subscript[π, M/K])]), M/K, \(⟶\&\(T\&~\)\), I, (Overscript[⟶, (Subscript[ι, I/N])]), N}, { , (Underscript[↖, (Subscript[ι, L/M])]), ↓(Subscript[i, L]), (Overscript[↗, T'])(Underscript[↙, S']),  ,  ,  }, { ,  , L,  ,  ,  ,  }}), T∶, ({{ ,  ,  ,  ,  ,  , N/J}, { ,  ,  ,  ,  , (Subscript[↙, (Subscript[i, I])]), (Subscript[↑, (Subscript[π, N/J])])}, {M, (Overscript[⟶, (Subscript[π, M/K])]), M/K, \(⟶\&\(T\&~\)\), I, (Overscript[⟶, (Subscript[ι, I/N])]), N}, { ,  ,  , ↘, (Underscript[⟶, (Subscript[T, ↓])]), ↗,  }})}, {1 - P∶, ({{M, (Overscript[⟶, (Subscript[π, M/K])]), M/
                  K, (Overscript[⟶, (Subscript[i, L])]), L, (Overscript[⟶, (Subscript[ι, L/M])]), M}, { ,  ,  , ↘, (Underscript[⟶, (Subscript[ι, L])]), ↗,  }}), Q∶, ({{ , ↗, (Overscript[⟶, (Subscript[π, J])]), ↘,  ,  ,  }, {N, (Overscript[⟶, (Subscript[π, N/J])]), N/
                    J, (Overscript[⟶, (Subscript[i, I])]), I, (Overscript[⟶, (Subscript[ι, I/N])]), N}})}})

Refinements and the Moore–Penrose Inverse

If both conditions are fulfulled, we may combine the effects of R and S, thus obtaining what is called the Moore–Penrose inverse (with respect to the chosen complements). As we will see soon, they are in bijective correspondence with the two–sided refinements already mentioned before.

From the above relations, one can immediately infer the alternative representations

\(T\^† = \(Sπ\_J = \(ι\_LR = 
          ST\&~\(\(R\)\(\ \)\(.\)\)\)\)\)

From the symbolic computation point of view, the Moore–Penrose inverse is interesting because it can also be described in terms of equations alone. Seen in a suitable (noncommutative) polynomial ring, these are polynomial equations and therefore in principle amenable to techniques like Gröbner bases. We will come back to this point later.

({{T(T^†)T = T}, {(T^†)T(T^†) = 
            (T^†)}, {(T^†)T = 1 - P}, {T(T^†) = Q}})

\(UT\_\(\(↓\)\(K\)\)π\_\(M/K\) = ι\_Lπ\_\(M/K\)\)

\(ι\_\(I/N\)T\^\(\(↑\)\(I\)\)U = ι\_\(I/N\)π\_J\)

As a consequence of this unique characterization, we obtain the following bijective correspondence: Each Moore–Penrose inverse (T^†) corresponds to exactly one choice of projectors P and Q. The Moore–Penrose characterization can therefore be added into the above list of correspondences—of course only if both projectors P and Q exist. In this sense, (T^†) encodes simultaneously the first group of data (kernel complements, sections, kernel projectors) and the second group (image complements, retractions, image projectors). Let us conclude by showing how to obtain these data explicitly:

({{L = Im (T^†), S = 
            (T^†)(Subscript[ι, I/N]), P = 
            1 - (T^†)T}, {J = 
            Ker (T^†), \(R = π\_\(M/K\)T\^†\), Q = T(T^†)}})

({{\(T\^† = ι\_LT\&~\^\(-1\)π\_J\), \(T\^† = 
            ST\&~R\), \(T\^† = \((1 - P)\)\_↓T\&~\^\(-1\)Q\^↑\)}})

Using the Moore–Penrose inverse, it is now easy to construct the two–sided refinement of a homomorphism along with its inverse.

\(T\&_\^\(-1\)T\&_⟨x, 
          y\_∞⟩ = \(T\&_\^\(-1\)⟨Tx + 
              y\_∞, 
            Px⟩ = \(\(⟨T\^†\((Tx + 
                    y\_∞)\) + 
              Px, \((1 - Q)\) \((Tx + 
                  y\_∞)\)⟩\)\(.\)\)\)\)

\(T\&_\(T\&_\^\(-1\)\) ⟨y, 
          x\_0⟩ = \(T\&_⟨T\^†y + x\_0, 
            y - Qy⟩ = \(\(⟨T\((T\^†y + 
                    x\_0)\) + y - Qy, 
            P\((T\^†y + 
                  x\_0)\)⟩\)\(.\)\)\)\)

As expected, there is also a bijective correspondence between Moore–Penrose inverses and refinements.

\(T\&_⟨x, 
          y\_∞⟩ = ⟨Tx + y\_∞, 
        Px⟩\)
\(T\&_\^\(-1\)⟨y, 
          x\_0⟩ = ⟨T\^†y + x\_0, 
        y - Qy⟩\)

\(T = π\_KT\&_ι\_M\)
\(T\^† = π\_MT\&_\^\(-1\)ι\_N\)

\(\(\(π\_NT\&_ι\_Mx = \(π\_NT\&_⟨x, 
            0⟩ = \(π\_N⟨Tx, 
              Px⟩ = 
          Tx\)\)\)\(,\)\)\)

\(\(\(π\_MT\&_\^\(-1\)ι\_Ny = \(π\_MT\&_\^\(-1\)⟨y, 
            0⟩ = \(π\_N⟨T\^†y, 
              y - Qy⟩ = 
          T\^†y\)\)\)\(,\)\)\)

Refinements of Endomorphisms

We can observe the same phenomenon even for the case of epimorphisms and monomorphisms. Indeed, if T ∶ M -> N is an epimorphism, its refinement is an operator \(TraditionalForm\`T\&_ ∶ M -> N×K\); if it is a monomorphism, the refinement comes as \(TraditionalForm\`T\&_ ∶ M×J -> N\). For an endomorphism T ∶ M -> M, this leads to the non–endomorphisms \(TraditionalForm\`T\&_ ∶ M -> M×K\) and \(TraditionalForm\`T\&_ ∶ M×J -> M\), respectively.
As before, let us start with the case of an epimorphism T ∶ M -> M. Here the idea was to refine the codomain from M to M×K such that the "lost" kernel components are preserved under the refinement \(TraditionalForm\`T\&_ ∶ M -> M×K\), defined through \(TraditionalForm\`T\&_x = ⟨Tx, Px⟩\) with P ∶ M -> M being the projector along L onto K. According to Proposition , its inverse \(TraditionalForm\`T\&_\^\(-1\) ∶ M×K -> M\) is given by \(TraditionalForm\`T\&_\^\(-1\)⟨y, 
            x⟩ = Sy + x\), where S ∶ M -> M is the section corresponding to the kernel complement L; it coincides with the Moore–Penrose inverse (T^†) for the kernel projector P and the (only possible) image projector (Subscript[1, M]). In order to make the domain agree with the codomain, one can simply iterate the process of adjoining the kernel in the following way.

\(⟨x, x\_0, 
        x\_1, …⟩ ↦ ⟨Tx, Px, x\_0, 
        x\_1, …⟩\)

\(\(\(⟨y, x\_0, 
        x\_1, …⟩ ↦ ⟨Sy + x\_0, 
        x\_1, …⟩\)\(\ \)\(,\)\)\)

\(T\&_\^\(-1\)\(T\&_\) ⟨x, x\_0, 
          x\_1, …⟩ = \(T\&_\^\(-1\)⟨Tx, Px, 
            x\_0, x\_1, …⟩ = \(⟨STx + 
              Px, x\_0, 
            x\_1, …⟩ = ⟨x, 
            x\_0, x\_1, …⟩\)\)\)

\(T\&_\(T\&_\^\(-1\)\) ⟨y, x\_0, 
          x\_1, …⟩ = \(T\&_⟨Sy + x\_0, 
            x\_1, …⟩ = \(⟨TSy + Tx\_0, 
            PSy + Px\_0, 
            x\_1, …⟩ = ⟨y, 
            x\_0, x\_1, …⟩\)\)\)

The differential operator \(TraditionalForm\`∂\(\(∶\)\(\(C\^∞\)\([\)\(0, \(\(∞\)\([\)\(\(->\)\(\(C\^∞\)\([\)\(0, \(\(∞\)\([\)\)\)\)\)\)\)\)\)\) is a typical example where we can apply endomorphic codomain refinement. Let us write 𝔉 for \(TraditionalForm\`\(\(C\^∞\)\([\)\(0, \(\(∞\)\([\)\)\)\)\) and use the decomposition \(TraditionalForm\`𝔉 = ℭ\( + \& . \)ℨ\), where ℭ denotes the space of constant functions \(TraditionalForm\`{\(\([[\)\(c\)\(]]\)\) | 
          c ∈ ℂ}\) and ℨ the space of initialized functions {f ∈ 𝔉 | 
            f(0) = 0}; note that Ker ∂ = ℭ. Then the induced nullspace projector is given by f ↦ [[f(
          0)]], and the induced section by \(TraditionalForm\`∫\(\(∶\)\(f ↦ \(\([[\)\(∫\_0\%t\( f(
                  t)\) dt\)\(]]\)\)\)\)\). Let us write …' for ∂… and s instead of \(TraditionalForm\`∂\&_\). Embedding 𝔉 into \(TraditionalForm\`𝔉×ℭ\^\((∞)\)\) in the obvious way, we have then sf = 
        f' + s [[f(0)]]. Applying s once more, this yields (s^2)f = 
        sf' + (s^2)[[f(
              0)]] and hence \(TraditionalForm\`s\^2f = 
        f'' + s\ [[f' \((0)\)]] + \(s\^2\)[[f(0)]]\). As we will see later, both of these are instances of the fundamental formula of the Mikusi(Overscript[n, '])ski calculus.
Now let us turn to the case of a monomorphism T ∶ M -> M. Here the idea was to refine the domain from M to M×J such that the "missing" codomain elements can be handed through by the refinement \(TraditionalForm\`T\&_ ∶ M×J -> M\), defined by \(TraditionalForm\`T\&_⟨x, 
            y⟩ = Tx + y\). According to Proposition , its inverse \(TraditionalForm\`T\&_\^\(-1\) ∶ M -> M×J\) is given by \(TraditionalForm\`T\&_\^\(-1\)y = ⟨Ry, \((1 - Q)\) 
            y⟩\), where Q ∶ M -> M is the projector along J onto I and R ∶ M -> M is the corresponding retraction; it coincides with the Moore–Penrose inverse (T^†) for the (only possible) kernel projector (Subscript[0, M]) and the image projector Q. In order to make the domain agree with the codomain, we iterate this time the process of adjoining the image complement as follows.

\(⟨x, y\_0, 
        y\_1, …⟩ ↦ ⟨Tx + y\_0, 
        y\_1, …⟩\)

\(⟨y, y\_0, 
        y\_1, …⟩ ↦ ⟨Ry, \((1 - Q)\) y, y\_0, 
        y\_1, …⟩\)

\(T\&_\^\(-1\)\(T\&_\) ⟨x, y\_0, 
          y\_1, …⟩ = \(T\&_\^\(-1\)⟨Tx + y\_0, 
            y\_1, …⟩ = \(⟨RTx + 
              Ry\_0, \((1 - Q)\) 
                Tx + \((1 - Q)\) y\_0, y\_0, 
            y\_1, …⟩ = ⟨x, 
            y\_0, y\_1, …⟩\)\)\)

\(T\&_\(T\&_\^\(-1\)\) ⟨y, y\_0, 
          y\_1, …⟩ = \(T\&_⟨Ry, \((1 - Q)\) y, y\_0, 
            y\_1, …⟩ = \(⟨TRy + \((1 - Q)\) y, y\_0, 
            y\_1, …⟩ = ⟨y, 
            y\_0, y\_1, …⟩\)\)\)

The integral operator \(TraditionalForm\`∫\(\(∶\)\(𝔉 -> 𝔉\)\)\), defined as above is a typical example where one may employ endomorphic domain refinement. Since Im ∫ = ℨ, we may use ℭ again, this time as an image complement, with corresponding image projector  f ↦ 
        f - [[f(
            0)]] and retraction ∂. Let us now write l instead of \(TraditionalForm\`∫\&_\), and this time we embed 𝔉 and also (ℭ^((∞))) into \(TraditionalForm\`𝔉×ℭ\^\((∞)\)\) in the obvious way. We write (δ^((i))) for the i–th unit vector of \(TraditionalForm\`ℭ\^\((∞)\) ⊆ 𝔉×ℭ\^\((∞)\)\) with i = 0, 1, 2, …, abbreviating (δ^((0))) by δ and (δ^((1))) by δ', etc. Then we have \(TraditionalForm\`lδ = \(\([[\)\(1\)\(]]\)\)\) and \(TraditionalForm\`l\^2δ' = \(\([[\)\(1\)\(]]\)\)\). Hence we have now added the delta distribution along with its derivatives in a completely algebraic way.
The reader will have noticed that the refined differential and integral operators ended up having the same domain and the same codomain. This is no coincidence as we will see now. In fact, even more is true: the refined operators are mutual inverses!

\(\(\(\(R\&_\) ⟨f, f\_0, 
          f\_1, …⟩ = ⟨Rf, Pf, f\_0, 
        f\_1, …⟩\)\(\ \)\(,\)\)\)

\(R\&_\^\(-1\)⟨f, f\_0, 
          f\_1, …⟩ = \(\(⟨Sf + f\_0, 
          f\_1, …⟩\)\(\ \)\(.\)\)\)

\(\(\(\(S\&_\) ⟨f, f\_0, 
          f\_1, …⟩ = ⟨Sf + f\_0, 
        f\_1, …⟩\)\(\ \)\(,\)\)\)

\(S\&_\^\(-1\)⟨f, f\_0, 
          f\_1, …⟩ = \(\(⟨Rf, \((1 - Q)\) f, f\_0, 
          f\_1, …⟩\)\(\ \)\(.\)\)\)

With this insight, let us look again at the case \(TraditionalForm\`𝔉 = \(\(C\^∞\)\([\)\(0, \(\(∞\)\([\)\)\)\)\) with the cross section \(TraditionalForm\`⟨∂\(\(,\)\(∫\)\)⟩\). Now we know that the refinened differentiation s and the refined integration l are inverses of each other, so sl = ls = 
          (Subscript[1, 𝔉]). This sheds more light on what has happened in the refinement: We have combined the two "improvements" carried out separately in Proposition  and Proposition , namely the preservation of integration constants (restoring injectivity) and the addition of distributions as artificial derivatives (restoring surjectivity). Indeed, applying the fundamental formula sf = 
        f' + s [[f(0)]] to the special case \(TraditionalForm\`f = \(\([[\)\(1\)\(]]\)\)\), we obtain only s [[1]] = 
        s [[1]], so the new differentiation seems to get "stuck" on constant functions—at least it does not annihilate them anymore. On the other hand, we have seen that \(TraditionalForm\`lδ = \(\([[\)\(1\)\(]]\)\)\) for a suitable new object δ. Multiplying the latter identity by s yields δ = 
        s [[1]], so now we see why differentation gets stuck on \(TraditionalForm\`\(\([[\)\(1\)\(]]\)\)\) as we observed above—simply because it cannot be expressed in terms of 𝔉. Incidentally, there is also an analytic motivation why the delta distribution should be the derivative of the "constant" function \(TraditionalForm\`\(\([[\)\(1\)\(]]\)\)\). If we extend it to all of ℝ by setting it zero on ]-∞, 0[, it appears as the Heaviside function h. And the "derivative" of h is of course well–known to be δ, which can also be made precise by Schwarz's theory of distributions.
Let us now pass to the general case of homomorphisms. All we have to do then is to combine the constructions used in Proposition  and Proposition .

\(TraditionalForm\`T\&_ ∶ 
      M×K\^\((∞)\)×J\^\((∞)\) -> 
        M×K\^\((∞)\)×
          J\^\((∞)\)\)

\(⟨x, x\_0, x\_1, …, y\_0, 
        y\_1, …⟩ ↦ ⟨Tx + y\_0, Px, x\_0, 
        x\_1, …, 
        y\_1, …⟩\)

\(\(\(⟨y, x\_0, x\_1, …, y\_0, 
        y\_1, …⟩ ↦ ⟨T\^†y + x\_0, 
        x\_1, …, \((1 - Q)\) y, y\_0, 
        y\_1, …⟩\)\(\ \)\(,\)\)\)

\(T\&_\^\(-1\)\(T\&_\) ⟨x, x\_0, 
          x\_1, …, y\_0, 
          y\_1, …⟩ = \(T\&_\^\(-1\)⟨Tx + y\_0, 
            Px, x\_0, x\_1, …, 
            y\_1, …⟩ = \(⟨T\^†Tx + 
              T\^†y\_0 + Px, x\_0, 
            x\_1, …, \((1 - Q)\) Tx + \((1 - Q)\) 
                y\_0, y\_1, …⟩ = ⟨x, x\_0, x\_1, …, y\_0, 
            y\_1, …⟩\)\)\)

\(T\&_\(T\&_\^\(-1\)\) ⟨y, x\_0, 
          x\_1, …, y\_0, 
          y\_1, …⟩ = \(T\&_⟨T\^†y + x\_0, 
            x\_1, …, \((1 - Q)\) y, y\_0, 
            y\_1, …⟩ = \(⟨TT\^†y + 
              Tx\_0 + \((1 - Q)\) y, 
            PT\^†y + 
              Px\_0, x\_1, …, y\_0, 
            y\_1, …⟩ = ⟨y, 
            x\_0, x\_1, …, y\_0, 
            y\_1, …⟩\)\)\)

As an example of such a general refinement, consider again \(TraditionalForm\`𝔉 = \(\(C\^∞\)\([\)\(0, \(\(∞\)\([\)\)\)\)\), but this time with the Eulerian differential operator \(TraditionalForm\`ϑ ∶ 𝔉 -> 𝔉\) defined by ϑu (t) = 
        t u'(t) as on page 219 in [20]. In this case, we have Ker ϑ = ℭ as before, so ϑ is non–injective. But ϑ is also non–surjective; in fact, Im ϑ = ℨ as we can see easily: Since every f ∈ Im ϑ can be written in the form f(t) = t u'(t), the condition f(0) = 0 is clearly necessary. Conversely, if this condition is fulfilled, we have [[f(t)/
          t]] ∈ 𝔉 since it is clearly smooth for t > 0 and approaches f' (0) for \(TraditionalForm\`t -> \(0\^+\)\) by l'Hospital's rule; hence the differential equation f(t) = t u'(t) has an integral  λf := 
        u ∈ 𝔉 with \(TraditionalForm\`u(
          t) = ∫\_0\%t\( 
                f(τ)\)/τ\ \ dτ\), where we have introduced the operator \(TraditionalForm\`λ ∶ ℨ -> 𝔉\). Note that we can restrict both ϑ and λ to operators \(TraditionalForm\`ϑ\_0, λ\_0 ∶ ℨ -> ℨ\), which are then mutually inverse. The refinement

\(ϑ\&_ ∶ 𝔉×ℭ\^\((∞)\)×ℭ\^\((∞)\) -> 𝔉×ℭ\^\((∞)\)×ℭ\^\((∞)\)\)

\(λ\&_\ [[
        t\ \(δ \((t)\)\)]] = \(\(\([[\)\(∫\_0\%t\(\( τ\ \(δ \((τ)\)\)\)\/τ\) dτ\)\(]]\)\) = \(\(\([[\)\(∫\_0\%t\( δ \((τ)\)\) dτ\)\(]]\)\) = \(\([[\)\(1\)\(]]\)\)\)\)\)

\(ϑ\&_\ [[∫t\^\(-1\)]] = \(\(\([[\)\(t\ \((∫t\^\(-1\))\)'\)\(]]\)\) = \(\(\([[\)\(t\ t\^\(-1\)\)\(]]\)\) = \(\([[\)\(1\)\(]]\)\)\)\)\)

Summary and Outlook

In the sequel, both approaches will turn out to be useful, but we will need them almost exclusively for restoring injectivity. The approach via the codomain refinement will lead to the Mikusi(Overscript[n, '])ski calculus in the next chapter, whereas the alternative via the Moore–Penrose inverse will give rise to what we shall accordingly call the Moore–Penrose calculus. Both approaches have their pros and cons as we will see.

The Mikusi(Overscript[n, '])ski Calculus

The Heaviside Calculus as a Precursor

For that, let us recall some basic concepts of analysis. We call a function \(TraditionalForm\`f ∶ ℝ -> ℂ\) locally integrable if it is absolutely integrable on each finite interval [a, b], i.e. if (∫_a^b)|f(
              t) dt| is finite for all a, b ∈ ℝ. Note that we must interpret ∫ in the sense of the Lebesgue integral. A locally integrable function may have (even infinitely many) discontinuities, as long as they cover only a zero set (a set with vanishing Lebesgue measure). Furthermore, we regard two functions as equal iff they differ only an a zero set, which is tantamount to saying f = g iff (∫_a^b) f(
              t) dt = (∫_a^b) g(
              t) dt for all a, b ∈ ℝ.
Including discontinuous functions into our considerations is not just a luxury of generality; it will turn out to be vital for the symbolic machinery. One such (mildly) discontinous function is the highly important Heaviside function (also called the unit–step function for obvious reasons), defined as h(t) = 0 for t < 0 and h(t) = 1 for t > 0. Note that we need not define h(t) at the point t = 0. For once, a function remains the same if we modify finitely many values (or even infinitely many values on an arbitrary zero set); and then h will remain discontinous no matter which value we would attach to t = 0. Nevertheless, some people set h(0) to 0, others to 1, still others to (1/2) in order to give it a symmetric appearance. We will not need to make any commitments of this kind.

The most important operator on 𝔏 will be integration, considered as the antiderivative with vanishing integration constant.

\(\(($f)\) \((t)\) = ∫\_0\%t\( 
            f \((τ)\)\) \(\(dτ\)\(\ \)\(.\)\)\)

Before proceeding to differentiation, let us first investigate the operator $ a bit further. Among its most crucial properties is the fact that an iterated integration may always be replaced by a single integration, provided one uses an appropriate integral kernel: In general, a kernel operator \(TraditionalForm\`u ∶ 𝔏 -> 𝔏\) is given by \(TraditionalForm\`\((uf)\) \((t)\) = ∫\_0\%t\( k(
              t, τ)\) \(f(τ)\) dτ\) for all f ∈ 𝔏, where \(TraditionalForm\`k ∶ \([0, \(\(∞\)\([\^2\)\(\(->\)\(ℂ\)\)\)\)\) is a suitably smooth kernel function. In this sense, iterated integration can be represented as a kernel operator.

\(\(($\^nf)\) \((t)\) = \(1\/\(\((n - 
                  1)\)!\)\) \(∫\_0\%t\(\((t - τ)\)\^\(n - 
                  1\)\) \(f \((τ)\)\) dτ\)\)

\(\(\(\(($\^\(n + 1\)f)\) \((t)\) = \(\(1\/\(\((n - 
                    1)\)!\)\) \(∫\_0\%t\(∫\_0\%σ\(\((σ - τ)\)\^\(n - 
                      1\)\) \(f \((τ)\)\) dτ dσ\)\) = \(1\/\(\((n - 1)\)!\)\) \(∫\_0\%t\(∫\_τ\%t\(\((σ - τ)\)\^\(n - 
                      1\)\) \(f \((τ)\)\) dσ dτ\)\)\)\)\(,\)\)\)

\(\(\(\(($\^\(n + 1\)f)\) \((t)\) = \(\(1\/\(\((n - 
                    1)\)!\)\) \(∫\_0\%t\( 
                f \((τ)\)\) \((∫\_τ\%t\(\((σ - τ)\)\^\(n - 
                          1\)\) dσ)\) dτ\) = \(\(1\/\(\((n - 
                      1)\)!\)\) \(∫\_0\%t\(\(\( 
                        f \((τ)\)\)\ [\((σ - τ)\)\^n\/n]\)\_\(σ = τ\)\%t\) dτ\) = \(1\/\(n!\)\) \(∫\_0\%t\( 
                  f \((τ)\)\)\ \(\((t - τ)\)\^n\) dτ\)\)\)\)\(\ \)\(,\)\)\)

Here the kernel is given by k(
          t, τ) = (((t - τ))^(n - 1))/(n - 1)!, so it has the special form of a convolution kernel: This is a kernel that depends only on t - τ, so it can be written in the form k(t, τ) = g(t - τ) for some function \(TraditionalForm\`g ∶ \([0, \(\(∞\)\([\)\(\(->\)\(ℂ\)\)\)\)\); in our case, g is obviously given by g(s) = (s^(n - 1))/(n - 1)!. The importance of convolution kernels is that their corresponding kernel operators \(TraditionalForm\`u ∶ 𝔏 -> 𝔏\) act as convolution, so uf = g*f in the sense of the following definition.

\(\((f*g)\) \((t)\) = ∫\_0\%t\( 
            f \((τ)\)\) \(g \((t - τ)\)\) dτ\)

The representation of iterated integration as a (suitably normalized) convolution with the power function \(TraditionalForm\`\(\([[\)\(t\^n\)\(]]\)\)\) is already an indication of the important rôle played by the convolution in extending the Heaviside calculus to the Mikusi(Overscript[n, '])ski calculus. For the time being, however, we will ignore these connections and continue to see ($^n) as the iterate of the operator introduced in Definition . Let us just note an important special case of Proposition .

\(\(\(\($\^n\)\ [[1]] = \(\([[\)\(t\^n/\(n!\)\)\(]]\)\)\)\(\ \)\(,\)\)\)

For defining a suitable differentiation, we construct the endomorphic domain refinement of the monomorphism \(TraditionalForm\`$ ∶ 𝔏 -> 𝔏\). In order to apply Proposition , we have to specify a complement for the image of $. Let us write \(TraditionalForm\`𝔄 ≤ 𝔏\) for the subspace of all absolutely continuous functions and \(TraditionalForm\`ℨ ≤ 𝔄\) for the subspace of those absolutely continuous functions that pass through the origin (the "initialized" absolutely continuous functions). Using this terminology, the image of $ is ℨ, and the space ℭ of constant functions generate a decomposition \(TraditionalForm\`𝔄 = ℨ\( + \& . \)ℭ\). Now choose any complement 𝔇 of 𝔄, such that \(TraditionalForm\`𝔏 = 𝔄\( + \& . \)𝔇\); note that such a complement exists since 𝔏 is a vector space, but there is no canonical choice. At any rate, we can think of 𝔇 as the—or rather: a—space of (absolutely) discontinuous functions. Introducing the sum space \(TraditionalForm\`𝔘 = ℭ + 𝔇\) of "unintegrals", we get the decomposition \(TraditionalForm\`𝔏 = ℨ\( + \& . \)𝔘\) so that 𝔘 may be chosen as the required image complement.  Comparing with the simpler setting in Chapter , we see that here $ fails to be surjective for two reasons—first because any integral passes through the origin, and second because some functions are not smooth enough to be integrals; this is reflected by having two complementary components in the decomposition \(TraditionalForm\`𝔏 = ℨ\( + \& . \)ℭ\( + \& . \)𝔇\). Writing now 𝔏' for \(TraditionalForm\`𝔏×𝔘\^\((∞)\)\), we end up with the refinement of $ that we shall call \(TraditionalForm\`l ∶ 𝔏' -> 𝔏'\), together with an inverse refinement denoted by \(TraditionalForm\`s ∶ 𝔏' -> 𝔏'\). Again we think of 𝔏 as embedded into 𝔏', such that s is an extension of $. Picking up terminology from [18], we will refer to 𝔏' as the space hyperfunctions (even though we are dealing with a much smaller "function space" that does not contain any "operators").
We shall now introduce the operator ∂ of differentiation as the retraction of $ that corresponds to the chosen decomposition \(TraditionalForm\`𝔏 = ℨ\( + \& . \)𝔘\). This means that \(TraditionalForm\`∂\( | \_ℨ\)\) is the inverse of the isomorphism \(TraditionalForm\`$\^↑ ∶ 𝔏 -> ℨ\), whereas \(TraditionalForm\`∂\( | \_𝔘\)\) is the zero operator. For f ∈ ℨ, we have clearly ∂f = 
        sf, applying the usual identifications. Now assume f ∈ 𝔄. According to the decomposition \(TraditionalForm\`𝔄 = ℨ\( + \& . \)ℭ\), we write f = in f + ev f where \(TraditionalForm\`in ∶ 𝔄 -> ℨ\) and \(TraditionalForm\`ev ∶ 𝔄 -> ℭ\) are the corresponding left and right projectors f ↦ 
        f - [[f(
            0)]] and f ↦ [[f(
          0)]], called "initialization" and "evaluation", respectively. Then we have sf = 
        s (in f) + s (ev f). But s coincides with ∂ on ℨ, so we have \(TraditionalForm\`s\ \((in\ f)\) = \(∂\ \((in\ f)\) = ∂f\)\), the latter because the integration constant [[f(
        0)]] is annihilated by ∂. Hence we have \(TraditionalForm\`sf = ∂f + 
          s\ [[f(0)]]\). The extra term s [[f(
          0)]] present in this identity first occurs in the function \(TraditionalForm\`h = \(\([[\)\(1\)\(]]\)\)\); note that, in our setting, the "constant function" \(TraditionalForm\`\(\([[\)\(1\)\(]]\)\)\) really coincides with the Heaviside function h because of the convention introduced in Definition . Abbreviating the hyperfunction sh by δ, we can now write the above identity as \(TraditionalForm\`sf = ∂f + \(f(0)\) δ\), a formula that we have essentially already derived after Proposition .
The hyperfunction δ ∈ 𝔏' shows that 𝔏' is a proper extension of 𝔏 since we have lδ = h, whereas there is no "classical function" in 𝔏 whose indefinite integral would give h. As before, we write (δ^((n))) for the iterates \(TraditionalForm\`s\^nδ ∈ 𝔏'\) with n > 0. But not only ℭ creates new hyperfunctions in 𝔏'; the same applies of course also to 𝔇. For example, \(TraditionalForm\`s\ [[1/\@t]]\) is a hyperfunction that does not come from 𝔏 since the standard derivative [[-1/
          2 (Sqrt[ (t^3)])]] of \(TraditionalForm\`\(\([[\)\(1/\@t\)\(]]\)\)\) is not locally integrable; in the theory of distributions, \(TraditionalForm\`s\ [[1/\@t]]\) would be addressed as the pseudofunction -(1/2) Pf (1/(Sqrt[(t^3)])), where a pseudofunction Pf (f) operates on test functions φ through (∫_0^∞) f(
            t) (φ(t) - φ(
                0)) dt.
The identity \(TraditionalForm\`sf = ∂f + \(f(0)\) δ\) just derived can immediately be generalized to iterated differentiation.

\(\(\(s\^nf = 
      f\^\((n)\) + ∑\+\(i = 1\)\%n\(\( 
                  f\^\((n - 
                      i)\)\) \((0)\)\)\ \ δ\^\((i - 1)\)\)\(\ \)\(,\)\)\)

\(\(\(s\^nf' = \(f\^\((n + 1)\) + ∑\+\(i = 1\)\%n\(\( f\^\((n + 1 - i)\)\) \((0)\)\)\ \ δ\^\((i - 1)\) = 
        f\^\((n + 1)\) + ∑\+\(i = 1\)\%\(n + 1\)\(\(f\^\((n + 1 - 
                        i)\)\) \((0)\)\)\ \ δ\^\((i - 1)\) - \(f \((0)\)\)\ δ\^\((n)\)\)\)\(\ \)\(,\)\)\)

Actually, the fundamental formula may also be revealed as a disguise for Taylor's formula. In order to see this, just multiply the formula by (l^n), and apply the obvious relation

\(l\^nδ\^\((i - 1)\) = \(l\^ns\^ih = \(l\^\(n - i\)h = \(\([[\)\(t\^\(n - i\)/\(\((n - 
                  i)\)!\)\)\(]]\)\)\)\)\)

\(f = \(l\^nf\^\((n)\) + ∑\+\(i = 1\)\%n\(\( 
                    f\^\((n - i)\)\) \((0)\)\)\ \ [[
              t\^\(n - i\)/\(\((n - i)\)!\)]] = 
        l\^nf\^\((n)\) + ∑\+\(j = 0\)\%\(n - 1\)\(\(f\^\((j)\)\) \((0)\)\)\ \ [[
              t\^j/\(j!\)]]\)\)

\(\(\(f \((t)\) = ∑\+\(j = 0\)\%\(n - 1\)\(\(\(f\^\((j)\)\) \((0)\)\)\ \ t\^j\)\/\(j!\) + \(1\/\(\((n - 
                    1)\)!\)\) \(∫\_0\%t\(\((t - τ)\)\^\(n - 
                    1\)\) \(\(f\^\((n)\)\) \((τ)\)\) dτ\)\)\(\ \)\(,\)\)\)

It is high time for an example now.

({{u' (t) - 
              2  u(t) = (2  (t^2) - 
                  1) (e^((t^2))) ,}, {u(0) = 1 ,}})

\(\((s - 2)\) 
        u = \(\([[\)\(\((2  t\^2 - 
              1)\) e\^\(t\^2\)\)\(]]\)\) + δ\)

In order to work further on this example, we have to extend our domain of operators, which currently consists only of linear combinations of l and s, along with their iterates. In order to invert operators like s - 2 above, one needs integral operators with convolution kernels as we may recall from analysis. This motivates the construction of the general convolution product on 𝔏' that we will introduce in the following definition. For doing so, however, it is practical to rewrite the hyperfunctions in a slightly more practical fashion, emphasizing the linear structure inherent in 𝔏'.

\(\(\(sg = \(\($\&_\^\(-1\)\) ⟨g, 
            0, …⟩ = \(⟨∂g, \((1 - Q)\)g, 
            0, …⟩ = ⟨0, g, 
            0, …⟩\)\)\)\(\ \)\(,\)\)\)

In the above decomposition f = 
        (Subscript[f, 0]) + s(Subscript[f, 1]) + … + 
          (s^n)(Subscript[f, n]), we call (Subscript[f, 0]) the regular component and (Subscript[f, 1]), …, (Subscript[f, n]) the singular components of f.

\(\(\(\((f\_0 + sf\_1 + … + 
            s\^nf\_n)\)*\((g\_0 + 
            sg\_1 + … + 
            s\^mg\_m)\) = ∑\+\(i = 0\)\%n\(∑\+\(j = 0\)\%m s\^\(i + j\)\ \((f\_i*
                g\_j)\)\)\)\(\ \)\(,\)\)\)

It is immediately clear that the properties of associativity, commutativity and bilinearity familiar from the convolution on 𝔏 immediately carry over to 𝔏'. Another such property that we know from the convolution on 𝔏 is still true on 𝔏', and this observation will turn out to be crucial in the sequel: the interaction of convolution with differentiation.

\(\(\(s \((f*
            g)\) = \(∑\+\(i = 0\)\%n\(∑\+\(j = 0\)\%m s\^\(i + j + 1\)\ \((f\_i*g\_j)\)\) = \(\((sf\_0 + 
                s\^2f\_1 + … + 
                s\^\(n + 1\)f\_n)\)*\((g\_0 + 
                sg\_1 + … + 
                s\^mg\_m)\) = 
          sf*g\)\)\)\(\ \)\(,\)\)\)

For any k ∈ 𝔏', let us write \(TraditionalForm\`\(k\^*\)\) for the convolution operator 𝔏' -> 𝔏' defined by f ↦ k*f. Note that for k ∈ 𝔏, this yields a convolution kernel in the sense discussed before Definition , thus justifying our terminology.
As an immediate consequence of Lemma , we obtain the integral operator l as a convolution operator \(TraditionalForm\`\(h\^*\)\), just as in the familiar case of 𝔏.

h*f = h*((Subscript[f, 0]) + s(Subscript[f, 1]) + … + 
              (s^n)(Subscript[f, n])) = h*(Subscript[f, 0]) + 
            h*s(Subscript[f, 1]) + … + 
            h*(s^n)(Subscript[f, n]) = h*(Subscript[f, 0]) + 
              s (h*(Subscript[f, 1])) + … + (s^n) (h*
                    (Subscript[f, n])) = l(Subscript[f, 0 ]) + 
                sl(Subscript[f, 1]) + … + 
                (s^n)l(Subscript[f, n]) = l ((Subscript[f, 0]) + 
                      s(Subscript[f, 1]) + … + 
                      (s^n)(Subscript[f, n])) = 
                lf

Finally, we shall note that Heaviside's version of the Dirac distribution δ also behaves in the usual way, namely as the neutral element of convolution, i.e. the convolution operator \(TraditionalForm\`\(δ\^*\)\) acts as the identity operator on 𝔏'.

We are now in the position to describe the inverse of such operators like s - 2, which appeared in Example . As announced before, inverting operators of this form leads to convolution operators.

\(\(\(\(\((s - λ)\)\ [[e\^\(λt\)]]\^*\)f = \(\((s - λ)\) \((\(\([[\)\(e\^\(λt\)\)\(]]\)\)*
              f)\) = \(s\ \((\(\([[\)\(e\^\(λt\)\)\(]]\)\)*
                  f)\) - λ\ [[e\^\(λt\)]]*
              f = \(\((λ\ [[e\^\(λt\)]] + δ)\)*
                f - λ\ [[e\^\(λt\)]]*f = \(δ*f = 
              f\)\)\)\)\)\(\ \)\(,\)\)\)

With this result in our hands, let us now complete the solution of Example .

\(u = \(\((s - 
                2)\)\^\(-1\)\) \((\(\([[\)\(\((2  t\^2 - 
                  1)\) e\^\(t\^2\)\)\(]]\)\) + δ)\)\)

\(u \((t)\) = e\^\(2t\) + ∫\_0\%t\(\( 
                  e\^\(2t - 
                    2τ\)\) \((2τ\^2 - 
                  1)\)\) \(e\^\(τ\^2\)\) dτ\)

In other examples, we may have to invert a higher–order differential operator of the form (((s - λ))^(n + 1)), and actually this is all we need for the general case as we will see soon.

\(\(1\/\(\((n + 
                  1)\)!\)\) \(\(\((s - λ)\)\^\(n + 
                    2\)\)[[
              t\^\(n + 1\)e\^\(λt\)]]\^*\) = \(\(1\/\(n!\)\) \((s - λ)\) \(\(\(\((s - λ)\)\^\(n + 
                            1\)\)[[\(t\^n\) e\^\(λt\)]]\^*\)[[
                t]]\^*\) = \(\(\((s - λ)\)\ [[e\^\(λt\)]]\^*\) = \(\(1\_\(𝔏'\)\)\(\ \)\(.\)\)\)\)\)

\(\(\([[\)\(\(t\^n\) e\^\(λt\)\)\(]]\)\)*\(\([[\)\(e\^\(λt\)\)\(]]\)\) = \(\(\([[\)\(∫\_0\%t\( τ\^n\) \(e\^\(λτ\)\) \(e\^\(λ \((t - τ)\)\)\) dτ\)\(]]\)\) = \(\(\([[\)\(\(e\^\(λt\)\) \(∫\_0\%t\( τ\^n\) dτ\)\)\(]]\)\) = \(\(\(1\/\(n + 1\)\)[[\(t\^\(n + 
                      1\)\) e\^\(λt\)]]\)\(\ \)\(.\)\)\)\)\)

We conclude now the chapter on the Heaviside calculus by summarizing the essential features of the set of convolution operators \(TraditionalForm\`{\(f\^*\) | f ∈ 𝔏'}\), which we will denote by \(TraditionalForm\`\(𝔏\^*\)\). First of all, it is clear that they form a complex vector space. In fact, much more is true.

({{\(\(\(𝔏\^*\)×𝔏' -> 𝔏'\)\(\ \)\(,\)\)}, {\(\((\(f\^*\), g)\) ↦ \(f\^*\) g = 
            f*\(\(g\)\(\ \)\(.\)\)\)}})

The ambivalence of 𝔏', appearing both as a space of (hyper)functions and—in the isomorphic guise of \(TraditionalForm\`\(𝔏\^*\)\)—as a ring of operators on exactly the same space: this ambivalence will be used heavily in the Mikusi(Overscript[n, '])ski calculus to be discussed in the next chapter.

The Mikusi(Overscript[n, '])ski Calculus in the Narrow Sense

\(a\_1\/b\_1 ± 
        a\_2\/b\_2 = \(a\_1b\_2 ± a\_2b\_1\)\/\(b\_1b\_2\)\)

\(\(a\_1\/b\_1\) 
        a\_2\/b\_2 = \(a\_1a\_2\)\/\(b\_1b\_2\)\)

\(\(a\_1\/b\_1\)/\(a\_2\/b\_2\) = \(a\_1b\_2\)\/\(b\_1a\_2\)\)

The reason why we emphasized that D need not have a multiplicative unit is of course because 𝔏 really has none. For assume there were some δ such that δ*f = f for all f ∈ 𝔏. Obviously, this δ acts like the Dirac distribution then, so we should not expect it to be contained in 𝔏. And using a somewhat more detailed argument, e.g. on page 58 in [16], one can prove this statement rigorously: The convolution ring 𝔏 does not contain a unit element (in the language of Schwarz distributions, this means that δ is a singular distribution). But the other properties are fulfilled: As mentioned earlier, the convolution * is associative, bilinear, and commutative, thus making a commutative ring; the fact that 𝔏 does not contain any zerodivisors with respect to * is a much deeper result of analysis, which may be found e.g. on page 346 of Mikusi(Overscript[n, '])ski's own book [17]. Let us summarize this

Having the necessary prerequisites, the Mikusi(Overscript[n, '])ski field is easy enough to define; the big question is only how to interpret and make use of it. Following [16], we will write the elements of (𝔏^ ∼)  as a so–called convolution fraction f // g, where f ∈ 𝔏 and \(TraditionalForm\`g ∈ 𝔏\_#\); this avoids confusion with the pointwise division of a function f by an almost nowhere vanishing function g. However, we should mention that many books, like Mikusi(Overscript[n, '])ski's own [17], simply write fg for f*g and accordingly f/g for f // g; this is also very reasonable if one is solely devoted to the Mikusi(Overscript[n, '])ski field since there are no pointwise operations available in it. Note that a term like f*h // g should be parsed as (f*h) // g.
Now what is the significance of a convolution fraction f // g as above? In analogy to the case of ℚ, its immediate meaning is just this: "something" that becomes f when it undergoes convolution with g. On the one hand, we have noted above that all of 𝔏 is embedded into \(TraditionalForm\`𝔏\^~\), as for any fraction field of an integral domain; hence \(TraditionalForm\`𝔏\^~\) contains all the functions of 𝔏. But not only that, there are also hyperfunctions here: We have already seen that \(TraditionalForm\`𝔏\^~\) contains a unit element that we may denote by δ again and introduced through, say, δ = h // h. On the other hand, there are also plenty of operators in \(TraditionalForm\`𝔏\^~\), when seen in the right way. One example is our good old friend l, operating on a function f ∈ 𝔏 through the Heaviside function h, as lf = h*f. In the Mikusi(Overscript[n, '])ski calculus, there are no other operators than those given by the convolution of a fixed element from \(TraditionalForm\`𝔏\^~\); hence one usually identifies l with h. This is also true for the inverse operator s, which arises from convolution by δ // h, so identifying gives s = δ // h.
Altogether we see that \(TraditionalForm\`𝔏\^~\) is somewhat fuzzy as far as the distinction between (hyper)functions and operators is concerned, as Mikusi(Overscript[n, '])ski himself conceded on page 28 of [17]. The point is just that an element \(TraditionalForm\`f ∈ 𝔏\^~\) may be considered as the right factor in a convolution product g*f, thus being the hyperfunction acted upon by the convolution operator \(TraditionalForm\`\(g\^*\)\); it may as well be seen as the left factor in f*g, now itself appearing as the convolution operator \(TraditionalForm\`\(f\^*\)\) acting on the hyperfunction g. Therefore it is a matter of taste whether one calls its elements by the name "operators", as Mikusi(Overscript[n, '])ski did in [17], or by the name "hyperfunctions" as Yosida did in [18]. We will continue to speak of convolution fractions, as inspired by [16].
Although \(TraditionalForm\`𝔏\^~\) does not admit a pointwise product of functions, one may at least multiply by numbers from ℂ. This gives rise to so–called numerical operators. Multiplying a function by α ∈ ℂ is realized by the convolution with \(TraditionalForm\`αδ := \(\([[\)\(α\)\(]]\)\) // \(\([[\)\(1\)\(]]\)\)\), and again α may be identified with αδ since this mapping \(TraditionalForm\`ℂ -> 𝔏\^~\) is really a ring monomorphism. This means amounts to saying that in a convolution fraction, δ may be replaced by 1 or just omitted. But we must be cautious not to confuse numerical operators α with the convolution operators induced by the constant function \(TraditionalForm\`\(\([[\)\(α\)\(]]\)\) = αh\). For example, convolution with 1 is the identity operator whereas convolution with \(TraditionalForm\`\(\([[\)\(1\)\(]]\)\) = h\) is integration. The delicate point is that \(TraditionalForm\`\(\([[\)\(1\)\(]]\)\)\) is the multiplicative unit of 𝔏 under pointwise multiplication of functions, whereas 1 is the new multiplicative unit of \(TraditionalForm\`𝔏\^~\) under the convolution product, namely the Dirac distribution δ.
But now let us return to another good old friend that we know already from Proposition , namely the fundamental formula, this time in the Mikusi(Overscript[n, '])ski calculus:

We could now go on with reproducing the identities derived in the Heaviside calculus, e.g. the inversion of (((s - λ))^n) described in Proposition  and Proposition . It is more efficient, however, to state once and for all that such identities may immediately be translated "from Heaviside to Mikusi(Overscript[n, '])ski", due to the following isomorphism theorem.

ιf = 
      (Subscript[f, 0]) + s*(Subscript[f, 1]) + … + 
        (s^n)*(Subscript[f, n]) .

ι((s^(k + l))(
          (Subscript[f, k])*(Subscript[g, l]))) = ((s^k)*
            (Subscript[f, k]))*((s^l)*(Subscript[g, l])) .

(Subscript[f, 0]) + s*(Subscript[f, 1]) + … + 
        (s^n)*(Subscript[f, n]) = 0 .

\(f = 
      s*f\_1 + … + 
          s\^n*f\_n ∈ ι𝔘\^\((∞)\)∖\(\({0}\)\(\ \)\(.\)\)\)

By the preceding proposition, it is clear that everything that one can do in the Heaviside calculus—like solving initial value problems as we did in Example  and Example —can be done analogously in the Mikusi(Overscript[n, '])ski calculus. Hence we may ask ourselves whether there is anything really new in the Mikusi(Overscript[n, '])ski field. In other words: Is the monomorphism ι introduced in the proof of the preceding proposition maybe even an isomorphism? In fact, it turns out that almost all "practically occurring examples" can usually be done in the Heaviside calculus: the operators involved in them are s–terms, i.e. convolution operators induced by hyperfunctions from 𝔏'. One example of an operator that is not in the image of ι is given by ζ := δ // ε with \(TraditionalForm\`ε := \(\([[\)\(1/e\^\(1/t\^2\)\)\(]]\)\)\); by its definition, we have ζ*ε = δ, with the convolution of 𝔏. Assume now there were some z ∈ 𝔏' with ι(z) = ζ. Since ι is a homomorphism, this means that z*ε = δ, this time with the convolution of 𝔏'. Writing \(TraditionalForm\`z = 
        z\_0 + sz\_1 + …s\^nz\_n\) with (Subscript[z, 0]) ∈ 𝔏 and (Subscript[z, 1]), …, 
      (Subscript[z, n]) ∈ 𝔘, we obtain

δ = (Subscript[z, 0])*ε + 
            s ((Subscript[z, 1])*ε) + … + 
            (s^n) ((Subscript[z, n])*ε) = 
          (Subscript[z, 0])*ε + (Subscript[z, 1])*ε' + … + 
            (Subscript[z, n])*(ε^((n))) ;

The Mikusi(Overscript[n, '])ski calculus is a rather well–devloped and extensive body of mathematics, a nice blend between analysis and algebra. In the frame of this lecture, we cannot follow its further unfolding, which may be looked up e.g. in the beautiful exposition [17] by Mikusi(Overscript[n, '])ski himself, or in the more modern and concise treatment [18]. Our goal here was just to clarify its main concepts, in particular how it relates to the main idea of this lecture: the symbolic representation and treatment of analysis operators. Nevertheless, let us give a brief account on the next steps typically advanced in building up the Mikusi(Overscript[n, '])ski calculus:

Notes on the Relation to the Laplace Transformation

\(∫\+0\%∞\( e\^\(\(-p\)t\)\) \(f(
          t)\) dt\)

\(f(t) = \(1\/\(2  π\ i\)\) \(∫\+\(q - i\ ∞\)\%\(q + i\ ∞\)\(e\^\(pt\)\) \(\(f\&_\)(
              p)\) dp\)\)

There are numerous other interesting properties of the Laplace transformation (see the above references for proofs and for a more extensive treatment). Here we will only mention two that are of interest for us: the convolution theorem and the differentiation formula, as formulated in the following two propositions:

\(p\^n\(\(f\&_\)(p)\) = \(\(f\^\((n)\)\)\&_\)(
          p) + ∑\_\(i = 1\)\%n\(\( f\^\((n - i)\)\)(0)\) 
            p\^i\)

The similarity to the Heaviside / Mikusi(Overscript[n, '])ski calculus should be striking! Proposition  says that the convolution ring (Subscript[𝔏, 0]) is isomorphic to the Hadamard ring (Subscript[𝔏, 1]), which sounds very much like Proposition . (The pointwise product of functions is also called the Hadamard product, hence we call a ring of functions under this product by the name Hadamard ring.) As to Proposition , we are of course reminded of the fundamental formula given in Proposition , only that the latter did not need any considerations about what converges where (this can be taken as a kind of generic slogan: "Mikusi(Overscript[n, '])ski = Laplace minus convergence"). In the sequel, we will make these similarities clear and rigorous.
First of all, we will make the similarity even more perfect by extending the Laplace transformation to certain distributions (we may employ the same notation as for the classical transformation since they coincide on the classical domain as one can see easily).

From now on we will leave out the attribute "distributional" since the classical concepts are anyway contained in the generalized transformation. In particular, let us also note that Proposition  and Proposition  remain valid when extended to (Subscript[𝔏, 0']); the latter may also be recast into the simpler form \(TraditionalForm\`p\^n\(\(f\&_\)(
            p)\) = \(\(f\^\((n)\)\)\&_\)(p)\) when (f^((n))) is understood in the distributional sense—with the Dirac terms accounting for the (possible) jumps at 0. So the most important object new in (Subscript[𝔏, 0']) is obviously—once again—the Dirac distribution δ := h', this time really in the sense of Laurent Schwarz's theory. Since one can easily compute ℒ[h] (p) = (p^(-1)), we obtain ℒ[δ] = 1; here we can see another analogy to the Mikusi(Overscript[n, '])ski calculus—namely that δ acts as the multiplicative unit. Moreover, since we have ℒ[(δ^((n)))] (p) = 
        (p^n), Proposition  may be rewritten as

\(\(\(\(s\&_\^n\) 
        f\&_ = \(f\^\((n)\)\)\&_ + ∑\_\(i = 1\)\%n\(\( f\^\((n - i)\)\)(
              0)\) \(δ\^\((i - 1)\)\)\&_\)\(\ \)\(,\)\)\)

\(\(\(s\^nf = 
      f\^\((n)\) + ∑\_\(i = 1\)\%n\(\( 
                  f\^\((n - 
                      i)\)\) \((0)\)\)\ \ δ\^\((i - 1)\)\)\(\ \)\(,\)\)\)

ℒ[
        f // g] = ℒ[f]/ℒ[
          g] ,

Note that we cannot extend the Mikusi(Overscript[n, '])ski–Laplace transformation to all of \(TraditionalForm\`\(𝔏\^~\)\), just as we cannot extend the classical  (and also not the distributional) Laplace transformation to all of 𝔏. An example is provided by \(TraditionalForm\`\(\([[\)\(e\^\(t\^2\)\)\(]]\)\) // 
        1 ∈ \(𝔏\^~\)∖\(𝔏\_0\%~\)\) and \(TraditionalForm\`\(\([[\)\(e\^\(t\^2\)\)\(]]\)\) ∈ 𝔏∖𝔏\_0\). Note also that we have, in the sense of the identifications we introduced, the proper inclusions \(TraditionalForm\`𝔏\_0 < 𝔏\_\(0'\) < \(𝔏\_0\%~\)\) and hence \(TraditionalForm\`𝔏\_1 < 𝔏\_\(1'\) < \(𝔏\_1\%~\)\). In order to see that \(TraditionalForm\`𝔏\_0 ≤ 𝔏\_\(0'\)\) is really a proper inclusion, it is enough to call on δ as a witness; for establishing the other inclusion \(TraditionalForm\`𝔏\_\(0'\) ≤ \(𝔏\_0\%~\)\) as a real extension, we may refer to the counter example ζ = δ // ε after Lemma  since \(TraditionalForm\`ε = \(\([[\)\(1/e\^\(1/t\^2\)\)\(]]\)\)\) does have a Laplace transform: if we could write ζ as a distributional derivative (f^((n))) for some f ∈ (Subscript[𝔏, 0]), we would obtain \(TraditionalForm\`ℒ[ζ] = \(s\&_\^nℒ[f] = ℒ[s\^n*f]\)\) and hence, by the bijectivity of ℒ, also ζ = (s^n)*f, contrary to the fact that ζ is not contained in the image of ι as we have seen after Lemma .
Let us now summarize the Mikusi(Overscript[n, '])ski–Schwarz relationship between convolution fractions and Laplace transforms:

We should also emphasize, though, that both approaches have their pros and cons: For example, Schwarz's theory of distributions can operate on the full space of distributions for locally integrable functions on all of ℝ. Mikusi(Overscript[n, '])ski's convolution fractions, however, can only be extended (naturally) to have denominators locally integrable functions with, say, left–bounded support (with 𝔏 being the special case of functions with support bounded to the left by 0). On the other hand, there are of course plenty of examples, where Schwarz's theory fails because the "real" Laplace integrals do not converge, but Mikusi(Overscript[n, '])ski's "formal" Laplace integrals embodied in the convolution fractions still make perfect sense, often leading to a classical final result whose calculation would be much messier without the use of these artifical objects. Compare this with the use of complex numbers for accelerating the solution of real–number problems—in the words of Hadamard (who should also receive some credits here, beyond those for the pointwise product of functions): The shortest way from a problem in the reals to a solution in the reals is often through the complex numbers! Similar things could be said about numerous other "ideal" objects created by mathematicians throughout the centuries.

The Moore–Penrose Calculus

The approach described above works nicely for linear differential operators with constant coefficients, and it can also be adapted to certain operators with variable coefficients (see the references given in Chapter ). However, as soon as we consider a boundary value problem (BVP), the whole approach gets stuck. There is a deep reason for this: In the Mikusi(Overscript[n, '])ski calculus, the differential operator ∂ is replaced by the bijective operator s, which "remembers" its initial value according to the Fundamental Formula \(TraditionalForm\`sf = ∂f + f(0)\). By this definition, the iterates (s^k) come out as remembering the first k initial values—this is just the Fundamental Formula in its general form \(TraditionalForm\`s\^kf = ∂f + \(\(f\^\((n - 1)\)\)(
              0)\)\ δ - … - \(f(
              0)\)\ δ\^\((n - 1)\)\). Now imagine we want to solve the following prototypical BVP:

({{u'' = f}, {u(0) = u(1) = 0}})

As already announced at the end of Chapter , we will now depart from the refinement approach to our general task of "inverting linear differential operators" as we did in the Mikusi(Overscript[n, '])ski calculus. We will choose instead the pseudo–inverse approach, elaborating it into what we will call the Moore–Penrose calculus. The basic idea is again very simple: Any differential operator has a unique pseudo–inverse if we have chosen suitable kernel and image projectors. In many cases, the involved function spaces may be set up in such a way that the differential operator is surjective, thus making the image projector superfluous. In the sequel, we will think of all differential operators as acting on (C^∞)[0, 1], which does indeed make all linear differential operators surjective. If one wants to have more general solution concepts, e.g. weak or distributional solutions in the sense of [26], one may simply replace (C^∞)[0, 1] by (Subscript[C, 0]^(-∞))[0, 1], the space of all boundary–valued distributions—it turns out that then all the formulae we will derive here are still valid. The reason is that we will work exclusively on the operator level, and the relations we consider are not sensitive to the choice of the function space as long as we remain in the large class of "faithful models" as explained in Definition 38 of [28].

The Green's System

We have already mentioned in Chapter  that the Moore–Penrose calculus (as opposed to that of Mikusi(Overscript[n, '])ski) is based on noncommutative polynomials. The reason for this is that the Moore–Penrose inverse (T^†) of a differential operator T does not commute with T; otherwise it would be a full inverse after all. More specifically, Theorem  gives us T(T^†) = 1 and (T^†)T = 1 - P if we assume that T is surjective and P is a chosen projector onto the nullspace of T. Note that the other two Moore–Penrose equations are redundant in this case: As to T(T^†)T = 
        T, this is even true without any further assumptions on T, as remarked in the proof of Theorem ; as to \(TraditionalForm\`T\^†TT\^† = T\^†\), it follows from postmultiplying T(T^†) = 1 by (T^†).
In the Mikusi(Overscript[n, '])ski calculus, boundary operators were naturally embedded in the form of delta distributions, e.g. in the Fundamental Formula su = u' + u(0) δ. In the Moore–Penrose calculus, we have to adjoin suitable boundary operators explicitly. Thinking of the prototypical BVP described above, it is very plausible to add the two operators L ∶ u ↦ u(0) and R ∶ u ↦ u(1) for extracting the left and the right boundary value, respectively. Does this allow us to formulate "all" boundary conditions typically arising in the applications? According to [26], a generic "two–point BVP" has the following form (obviously one may use any other interval instead of the unit interval by a suitable scaling transformation): Given f ∈ (C^∞)[0, 1 ], find u ∈ (C^∞)[0, 1] such that Tu = f and (Subscript[B, 1])u = (Subscript[β, 1]), …, 
      (Subscript[B, n])u = (Subscript[β, n]) for a certain linear differential operator

Tu = 
      (Subscript[a, n])(u^((n))) + 
        (Subscript[a, n - 1])(u^((n - 1))) + … + (Subscript[a, 1]) 
          u' + (Subscript[a, 0])u

(Subscript[B, i]) u = (Subscript[p, i, 
              n - 1]) (u^((n - 
                  1))) (a)  + … + (Subscript[p, i, 1]) 
          u' (a) + (Subscript[p, i, 0]) u(
            a) + (Subscript[q, i, n - 1]) (u^((n - 1)))(
            b)  + … + (Subscript[q, i, 1]) 
          u' (b) + (Subscript[q, i, 0]) u(b)

T = ⌈(Subscript[a, n])⌉ 
          (D^n) + ⌈(Subscript[a, n - 1])⌉ 
          (D^(n - 1)) + … + ⌈(Subscript[a, 1])⌉ 
          D + ⌈(Subscript[a, 0])⌉

\(B\_i = 
      p\_\(i, n\)LD\^\(n - 1\)\  + … + p\_\(i, 1\)LD + 
        p\_\(i, 0\)L + 
        q\_\(i, n - 1\)RD\^\(n - 1\) + … + q\_\(i, 1\)RD + 
        q\_\(i, 0\)R\)

Up to now, we have considered BVPs that are inhomogeneous both in their differential equation and their boundary conditions. If f = 0, only the boundary conditions (obviously the easier part) remain inhomogeneous; such a BVP is called semi–homogeneous. If (Subscript[β, 1]), …, (Subscript[β, n]) = 0, only the differential equation (the "interesting" part) remains inhomogeneous; such a BVP is accordingly called a semi–inhomogeneous problem. From now on, we will restrict ourselves to problems of the latter type since they are already the general situation, in the following sense (see [26] for more details): For every "fully inhomogeneous problem", the solution u can be decomposed as (Subscript[u, h]) + (Subscript[u, i]), with (Subscript[u, h]) the solution of the corresponding semi–homogeneous problem and (Subscript[u, i]) is the solution of the semi–inhomogeneous problem. The computation of (Subscript[u, h]) is fairly straightforward (since we will anyway need a fundamental system for computing (Subscript[u, i])), hence the restriction to semi–inhomogeneous problems is no essential  loss of generality.
We have now introduced three "basic" operators D, L, R such that we can express any differential and boundary operators in terms of these. We will follow this strategy of building up complex operators from basic components also for the pseudo–inverses of the differential operators. Without further thinking, it is clear that the (pseudo)inverse of a differential operator must involve integration somehow. Hence we shall adjoin the "antiderivative operator" A, defined as follows:

\(Au = \(\([[\)\(∫\_0\%t\( 
            u \((τ)\)\) dτ\)\(]]\)\)\)

\(Bu = \(\([[\)\(∫\_t\%1\( 
            u \((τ)\)\) dτ\)\(]]\)\)\)

The deeper reason why we need both A and B but nothing else becomes clearer when we introduce the so–called Green's function of a two–point BVP in the form above. This is a function \(TraditionalForm\`g ∶ \(C\^∞\)[0, 1]\^2 -> \(C\^∞\)[0, 1]\) such that the "Green's integral"

\(u = \(\([[\)\(∫\_0\%1\( 
            g \((t, τ)\)\) \(f \((τ)\)\) dτ\)\(]]\)\)\)

We have now essentially introduced a noncommutative polynomial ring in the "indetermines" D, L, R, A, B. Strictly speaking, we must also count each multiplication operator ⌈a⌉ for some a ∈ (C^∞)[0, 1] as one indeterminate. The latter is quite uneconomical for at least two reasons: First of all, we usually do not need one multiplication operator for every element of the full ℂ–algebra (C^∞)[0, 1]. The coefficient functions (Subscript[a, 0]), …, (Subscript[a, n]) generate a small subalgebra 𝔄 of (C^∞)[0, 1] that may be conveniently enlarged a bit so to fit the fundamental system of the corresponding homogeneous equation Tu = 0. Second, we may always (since 𝔄 is a fortiori a complex vector space!) pick some basis \(TraditionalForm\`𝔄\^#\) in the resulting multiplier algebra 𝔄 ≤ (C^∞)[0, 
          1]. This has the effect that every a ∈ 𝔄 has a unique representation \(TraditionalForm\`α\_1b\_1 + … + α\_nb\_n\) with complex coefficients (Subscript[α, 1]), …, (Subscript[α, n]) and basis elements \(TraditionalForm\`b\_1, …, 
      b\_n ∈ 𝔄\^#\), such that ⌈a⌉ may be written more canonically in the form \(TraditionalForm\`\(α\_1\) ⌈b\_1⌉ + … + \(α\_n\) ⌈b\_n⌉\). As a consequence, it is sufficient to adjoin only the multiplication operators ⌈b⌉ induced by basis elements \(TraditionalForm\`b ∈ 𝔄\^#\). Note that many multiplier algebras occurring in applications come with a natural basis, i.e. are even defined in terms of them. For example, if one deals with linear differential operators with constant coefficients (as we will mostly do in the rest of these lecture notes), it is known from the theory that all the basis solutions (the basis elements in a fundamental system) may be expressed as so–called exponential polynomials: this is the complex algebra \(TraditionalForm\`𝔈𝔵𝔭\) built on the basis \(TraditionalForm\`𝔈𝔵𝔭\^# = {\(\([[\)\(x\^ke\^\(λt\)\)\(]]\)\) | 
            
            k ∈ ℕ ∧ λ ∈ ℂ}\).

\(ℂ⟨{D, 
            A, B, L, 
            R} ⋃ {⌈a⌉ | 
              a ∈ 𝔄\^#}⟩\)

The notation 𝒜𝓃(𝔄) comes from the older terminology "analytic polynomials". The idea is that integration and differentiation make up the core of analysis, hence the name. In order to make D really behave like a differential operator and A, B like integral operators, we have to breathe life into these symbols—up to now, D as well as A and B are nothing else than naked indeterminates of the polynomial ring 𝒜𝓃(𝔄). If we want to capture the spirit of analysis into the integro–differential polynomials, the first thing that comes to mind is the fundamental theorem of calculus, which states that integration is the pseudo–inverse of differentiation. More specifically, we get DA = 1 and DB = -1 as the operator analogs for the familiar theorem (∂_t)(∫_a^t) 
                f(τ) dτ = 
        f(t); furthermore we have AD = 1 - L and BD = R - 1 as the operator versions of (∫_a^t) 
              f' (τ) dτ = 
        f(t) - f(a). Once again we see the noncommutative nature of the Moore–Penrose calculus, this time on the level of the basic operators: Instead of just having a collection of commuting basic operators (Subscript[H, 1]), …, (Subscript[H, r]) with inverses (Subscript[H, 1]^(-1)), …, (Subscript[H, r]^(-1)) commuting with each other as well as with all the (Subscript[H, 1]), …, (Subscript[H, r]), we are led to a calculus that has to introduce both the basic operators (Subscript[H, 1]), …, (Subscript[H, r]) and (some of) its pseudo–inverses, say (Subscript[H, r + 1]), …, (Subscript[H, r + s]), along with their characteristic interaction relations (Subscript[H, i])(Subscript[H, j]) = 
        F((Subscript[H, 1]), …, (Subscript[H, r + s])) for all i = 1, …, r + s and j = 1, …, i - 1. In fact, an interaction may have a somewhat more general form, involving a slightly more complicated left–hand side (but most of the interactions are of the simple type displayed).
Deriving the relevant interaction relations needs some insight into the problem—in particular for judging whether one has indeed found "all the necessary interactions". Since it is too tedious to describe all the details here, let us just present the result of this analysis in the formulae given below, which we have called the Green's system (for reasons that will become obvious soon). Most of the interactions there are fairly straighforward, e.g. the famous "Leibniz law" \(TraditionalForm\`D\ ⌈a⌉ = ⌈a⌉ D + ⌈a'⌉\) giving an operator form to the more familiar product rule (au)' = 
        au' + a'u. What is perhaps a bit more surprising is the following asymmetry between differentiation and integration: While iterated differentiation can never by reduced to a single differentation, an iterated integral may be rewritten as a single integral with a suitable kernel as we have seen this in Proposition ; this is the contents of the equation labelled "AA". Using integration by parts, this may be easily extended to double integrations of the form A ⌈a⌉ A as described in "AMA"; hence it is true for any succession of integrals and multipliers.

System["Equalities for Algebraic Simplication",any[a, b],
({{⌈a⌉ ⌈b⌉ = ⌈a b⌉, "MM"}})
]

System["1. Equalities for Isolating Differential Operators",any[a],
({{D A = 1, "DA"}, {D B = -1, "DB"}, {D ⌈a⌉ = ⌈a⌉ D + ⌈a'⌉, "DM"}, {D L = 0, "DL"}, {D R = 0, "DR"}})
]

System["2. Equalities for Isolating Boundary Operators",any[a],
({{L A = 0, "LA"}, {R A = A + B, "RA"}, {L B = A + B, "LB"}, {R B = 0, "RB"}, {L ⌈a⌉ = (a^ ←)  L, "LM"}, {R ⌈a⌉ = (a^ →)  
                  R, "RM"}, {L L = L, "LL"}, {L R = R, "LR"}, {R L = L, "RL"}, {R R = R, "RR"}})
]

System["3. Equalities for Contracting Integration Operators",any[a],
({{\(A\ ⌈a⌉\ A = ⌈\(∫\^*\)f⌉ A - 
                  A\ ⌈\(∫\^*\)f⌉\), "AMA"}, {\(A\ ⌈a⌉\ B = ⌈\(∫\^*\)f⌉ B + 
                  A\ ⌈\(∫\^*\)f⌉\), "AMB"}, {\(B\ ⌈a⌉\ A = ⌈\(∫\_*\)f⌉ A + 
                  B\ ⌈\(∫\_*\)f⌉\), "BMA"}, {\(B\ ⌈a⌉\ B = ⌈\(∫\_*\)f⌉ B - 
                  B\ ⌈\(∫\_*\)f⌉\), "BMB"}, {\(A\ A = ⌈\(∫\^*\)1⌉\ A - 
                  A\ ⌈\(∫\^*\)1⌉\), "AA"}, {\(A\ B = ⌈\(∫\^*\)1⌉\ B + 
                  A\ ⌈\(∫\^*\)1⌉\), "AB"}, {\(B\ A = ⌈\(∫\_*\)1⌉ A + 
                  B\ ⌈\(∫\_*\)1⌉\), "BA"}, {\(B\ B = ⌈\(∫\_*\)1⌉ B - 
                  B\ ⌈\(∫\_*\)1⌉\), "BB"}})
]

System["4. Equalities for Absorbing Integration Operators",any[a],
({{A ⌈a⌉ D = -(f^ ←)  L + ⌈a⌉ - 
                  A ⌈a'⌉, "AMD"}, {B ⌈a⌉ D = (f^ →)  
                    R - ⌈a⌉ - 
                  B ⌈a'⌉, "BMD"}, {A D = -L + 1, "AD"}, {B D = R - 1, "BD"}, {\(A ⌈a⌉ 
                  L = ⌈\(∫\^*\)a⌉ L\), "AML"}, {\(B ⌈a⌉ 
                  L = ⌈\(∫\_*\)a⌉ L\), "BML"}, {\(A ⌈a⌉ 
                  R = ⌈\(∫\^*\)a⌉ R\), "AMR"}, {\(B ⌈a⌉ 
                  R = ⌈\(∫\_*\)a⌉ R\), "BMR"}, {\(A\ L = ⌈\(∫\^*\)a⌉\ L\), "AL"}, {\(B\ L = ⌈\(∫\_*\)a⌉\ L\), "BL"}, {\(A\ R = ⌈\(∫\^*\)a⌉\ R\), "AR"}, {\(B\ R = ⌈\(∫\_*\)a⌉\ R\), "BR"}})
]

The above formulae are verbatim Theorema input, and the quantifiers over a and b are supposed to range over the algebra basis \(TraditionalForm\`𝔄\^#\). They involve certain abbreviations that we will also use in the sequel; their key idea is that all the basic operators have natural analogs on the function level embodied by 𝔄: As to the differential operator D, we continue to use u ↦ u' as its functional analog. For the integral operators A and B, we have introduced the notation \(TraditionalForm\`u ↦ \(∫\_*\)u\) and \(TraditionalForm\`u ↦ \(∫\^*\)u\), respectively. For the boundary operators L and R, the notations u ↦ (u^←) and u ↦ (u^ ->)  are used, respectively. Finally, the multipliers ⌈a⌉ obviously do not need any analogs since multiplication is a "native" operation in the algebra 𝔄.
Note that all the equations labelled "xMy" subsume those labelled "xy"; we have nevertheless retained the latter since they give rise to reductions of polynomials with different terms: For example, the polynomial AA is different from any polynomial of the form A ⌈a⌉ A since the latter cannot have \(TraditionalForm\`a = \(\([[\)\(1\)\(]]\)\)\); this would need a spurious basis element \(TraditionalForm\`\(\([[\)\(1\)\(]]\)\)\). Speaking of basis elements, we have to mention another subtlety: Whenever the right–hand side of the above equations involves operations of 𝔄, the corresponding multipliers have to be understood in an expanded sense: As explained above, any non–basis multiplier of the form \(TraditionalForm\`⌈α\_1b\_1 + … + α\_nb\_n⌉\) must be replaced by \(TraditionalForm\`\(α\_1\) ⌈b\_1⌉ + … + \(α\_n\) ⌈b\_n⌉\), and the corresponding polynomial must be expanded; these steps have to be carried out until there are no more non–basis multipliers (obviously this process will always terminate).

The Green's Algebra as a Model of Operators

The Green's system has three crucial properties that will turn out to be important for the process of solving BVPs in the form specified above. First of all, the rewrite system generated from it is confluent, which means that all reductions starting from a given polynomial will come together (Latin "confluere") eventually. Algebraically speaking, this means just that the corresponding set of polynomials (obtained from the left–hand sides after rewriting each identity of the Green's system in the form … = 0) forms a noncommutative Gröbner basis of the ideal generated by it (called the "Green's ideal" in [28]), in the sense of Theorem 8 in [29]. Note, however, that this set of polynomials is infinite since each identity involving multipliers is actually an infinite family of identities, one for every basis element of 𝔄 (still we may speak of a finitary set since its description is in finite terms). As in the commutative case, it is enough to prove that all overlap ambiguities of the reduction system are resolvable; see Lemma 1.2 of [30]. We can do this in the usual manner by showing that the S–polynomial (Subscript[w, 2])(Subscript[p, 1]) - 
        (Subscript[p, 2])(Subscript[w, 1]) reduces to 0 for any pair of rules w(Subscript[w, 1]) = (Subscript[p, 1]) and (Subscript[w, 2])w = (Subscript[p, 2]).
It turns out that there are 233 S–polynomials to be considered, and the task of doing all these reductions is rather dull. Therefore we have automated this proof in the TH∃ OREM∀  system. This illustrates also a general principle of the TH∃ OREM∀  philosophy: any good "mathematical assistant" system should provide support an all levels of mathematics—here the "object level" of computation that uses the Green's system for solving BVPs in a way that will be illustrated soon; and the "meta level" of proving certain properties like confluence here. For more on the principles of object and meta level, see [31]. Since the whole automated proof contains approximately 2000 lines, let us just show some typical lines below. In every intermediate expression, the redex is framed by the system in order to improve readability. The uppercase letters above the equality symbol refer to the corresponding identity of the Green's system; the lowercase letters refer to certain identities on the functional level of 𝔄. The symbol ∮ is an abbreviation for the "definite integral" \(TraditionalForm\`\(∫\^*\)\(+\(∫\_*\)\)\).

⋮

⋮

Second, the Green's system is also noetherian, meaning that all reductions terminate—as opposed to the commutative case (Dixon's lemma!), this is not true for general polynomial sets. We have given two different proofs for this in [28]. The more intuitive proof uses the idea of various termination measures associated with the identities. For example, several identities decrease the "differential weight" (the number of occurrences of the indeterminate D), wheras none of the identities increases it. The other proof proceeds on a more algebraic line: A suitable graded lexicographic ordering is set up on the word monoid \(TraditionalForm\`\(Ω\^*\)\) over Ω = {D, A, B, L, R, M}, which is then extended to a well–ordering on the system of finite subsets of \(TraditionalForm\`\(Ω\^*\)\). This well–ordering induces a noetherian strict partial order on 𝒜𝓃(𝔄) by identifying all ⌈a⌉ with M and taking the support of the resulting polynomial. Hence it suffices to prove that all reductions are compatible with the induced order—which is easily achieved.
Confluence and noetherianity together imply that all reductions end in a unique normal form: The latter gives existence, whereas the former provides uniqueness. A rewrite system with these two properties is also called convergent.

Third, it is easy to see that all integral operators written in terms of a Green's function are in normal form; hence reducing an intermediate expression denoting the Green's operator will let us extract the Green's function easily. More precisely, we can characterize these normal forms by the following BNF grammar.

It will not come as a surprise now that these Green's polynomials are really the normal forms with respect to the Green's system of reductions (whose existence and uniqueness we have already ensured), hence our anticipated terminology is justified.

The proof of this theorem is rather straightforward, albeit slightly technical. It is easy to see that any Green's polynomial is indeed irreducible. For proving the converse, one takes an arbitrary monomial \(TraditionalForm\`p ∈ \(𝒜𝓃(𝔄)\)∖\(\(𝒜𝓃\_↓\)(𝔄)\)\) and shows that it is reducible, using a case distinction on the first letter of p. Despite its rather technical proof, the statement of the theorem is actually very intuitive: Any linear integro–differential–boundary operator must be a superposition of purely integral or differential or boundary operators (algebraic operators can be seen as zero–order differential operators). This is clear: on the monomial level, integration and differentiation cancel each other, and boundary evaluation collapses the functional range to a single point.
The above presentation of the normal forms as Green's polynomials allows us to immediately extract the Green's function from a Green's operator—hence the name. First of all, recall that the "differential weight" (the number of occurrences of D in an integro–differential polynomial) is invariant under the Green's system. Since the Green's operator is only formed from integrations, multipliers and boundary operators (as we shall see more precisely below), its normal form cannot be of type 𝒜𝒟 or 𝒜ℬ𝒟, hence must be of type 𝒜ℐ𝒜. So each monomial has the form ⌈f⌉ 
        A ⌈g⌉ or ⌈f⌉ 
        B ⌈g⌉, where f or g may also be 1; it contributes a term f(x) g(ξ) to the "upper" or "lower" part (the one with A to the upper, the one with B to the lower) of a Green's function defined by the case distinction

g(x, ξ)={({{upper(x, ξ), ⟸, a ≤ ξ ≤ 
                x ≤ b ,}, {lower(x, ξ), ⟸, a ≤ 
                x ≤ ξ ≤ b ,}})

We have now accumulated some properties of the integro–differential polynomials of 𝒜𝓃(𝔄) and their normal forms in \(TraditionalForm\`\(𝒜𝓃\_↓\)(𝔄)\). But what is their relationship to the "real" operators, e.g. the differential operator ∂∶(C^∞)[0, 
              1] -> (C^∞)[0, 1], modeled by \(TraditionalForm\`D ∈ \(𝒜𝓃\_↓\)(𝔄)\). In order to clarify this relationship, let us write \(TraditionalForm\`\(𝒜𝓃\_0\)(𝔄)\) for the set of all the polynomials l - r where l = r is any identity of the Green's system and the variables f and g in it range over all of \(TraditionalForm\`𝒜\^#\). The ideal generated by it, written as \(TraditionalForm\`⟨\(𝒜𝓃\_0\)(𝔄)⟩\), may be considered as the collection of different ways of writing the zero operator 0 ∶ (C^∞)[0, 
            1] -> (C^∞)[0, 1] by an integro–differential polynomial of 𝒜𝓃(𝔄); this is true because all the identities of the Green's system hold when its indeterminates are interpreted in the "obvious" way (e.g. D by ∂). Consequently, every representation of an operator T ∶ (C^∞)[0, 
            1] -> (C^∞)[0, 1] by some \(TraditionalForm\`p ∈ 𝒜𝓃(𝔄)\) has infinitely many equivalent representations p + z with \(TraditionalForm\`z ∈ ⟨\(𝒜𝓃\_0\)(𝔄)⟩\) a polynomial representing the zero operator 0 ∶ (C^∞)[0, 
            1] -> (C^∞)[0, 1]. Getting rid of these ambiguities, we may form the quotient algebra \(TraditionalForm\`\(𝒜𝓃(𝔄)\)/⟨\(𝒜𝓃\_0\)(𝔄)⟩\); its elements can then be considered as the "true" algebraic model of the actual operators (for a more rigorous treatment of the concept of model we refer to [28] and [33]). But as such, they are not feasible by symbolic computation—each equivalence class has infinitely many elements. And it is here that a crucial property of Gröbner bases comes to our aid. Let us recall what we have already mentioned above:

The crucial property of Gröbner bases in this context is that the normal forms with respect to the polynomial reduction induced by such a basis is a system of representatives for the corresponding quotient algebra, and the ring operations may be carried out on these representatives (with subsequent reduction). This is a very general fact about canonical simplifiers—see page 13 of [32]. In our present case, this means that \(TraditionalForm\`\(𝒜𝓃\_↓\)(𝔄)\) with its inherited operations forms an algebra isomorphic to the quotient algebra, with the decisive advantage of being computable.

As an algebra, the Green's polynomials \(TraditionalForm\`\(𝒜𝓃\_↓\)(𝔄)\) will be called the Green's algebra. In this sense, the Green's algebra represents the "algebraic essence" of those operators (C^∞)[0, 1] -> (C^∞)[
          0, 1] that we are interested in. This is particularly true about the integral operators among them (those of type 𝒜ℐ𝒜), which typically turn up as representing Green's operators (and actually can always be conceived as such).

Solving Boundary Value Problems

As mentioned above, the Green's function g can always be extracted from its normal–form Green's operator \(TraditionalForm\`G ∈ \(𝒜𝓃\_↓\)(𝔄)\). Since G is supposed to be the operator mapping f to u, we may substitute u = Gf into the problem, thus obtaining TGf = f and \(TraditionalForm\`B\_1Gf = \(… = \(B\_nGf = 0\)\)\) for all f ∈ (C^∞)[0, 1]. By definition of operator identity, this just means that TG = 1 and \(TraditionalForm\`B\_1G = \(… = \(B\_nG = 0\)\)\); with 1 and 0 being the identity and zero operator, respectively. Hence we have now reduced everything to a problem within \(TraditionalForm\`\(𝒜𝓃\_↓\)(𝔄)\). Just like any other system of polynomial equations, we might already stop here and hand it over to some computer algebra system capable of using noncommutative Gröbner bases for solving noncommutative polynomial systems. In fact, this is what we have done in [34], using the (rather) unreliable system NCAlgebra, developed at San Diego. There are a couple of disadvantages to this approach, though: First of all, it is very costly to compute a new Gröbner basis for each problem. Second, we have no a–priori guarantee that a finite such basis exists (unlike in the commutative case where such a guarantee can be given—apart from complexity concerns). Third, we would have to fuse the Green's system (which stays the same for all BVPs) and the above n + 1 equations (which represent the given BVP), a rather ugly mixture.
Hence we are looking for an approach that does better than this. The key is of course—as already announced—the Moore–Penrose inverse. As mentioned at the beginning of Section , the Moore–Penrose inverse of a differential operator T ∶ (C^∞)[0, 
            1] -> (C^∞)[0, 1] is characterized by the two equations T(T^†) = 1 and (T^†)T = 1 - P. The first of these suggests that G might be constructed as a Moore–Penrose inverse of T, provided we can find a suitable kernel projector P. The remarkable insight about these two equations is that the second of them is sufficient: it implies the first.

The lemma implies that constructing the Moore–Penrose inverse of T associated with a kernel projector P boils down to solving GT = 1 - P for G and that such a G will automatically fulfill TG = 1 then—the differential equation of the boundary value problem. Therefore we will try to search such a kernel projector P that the corresponding Moore–Penrose inverse G will also fulfill the given boundary conditions \(TraditionalForm\`B\_1G = \(… = \(B\_nG = 0\)\)\). It turns out that such a choice is always possible, and it needs only a bit of linear algebra to come up with the right operator. We will omit the technical details here—they may be looked up in [28]—and just present the result in one succinct formula for the kernel projector.
In order to do so, it will be convenient to introduce some matrix notation (we use overhat symbols for denoting vectors and matrices). Writing \(TraditionalForm\`\(D\&^\)\_n\) for the "Wronski operator" ⟨1, D, (D^2), …, 
        (D^(n - 1))⟩, the compound boundary operator ⟨(Subscript[B, 1]), …, 
        (Subscript[B, n])⟩ may be written as \(TraditionalForm\`\((L\(l\&^\) + 
            R\(r\&^\))\) \(D\&^\)\_n\) for suitable coefficent matrices \(TraditionalForm\`\(l\&^\), \(r\&^\) ∈ ℂ\^\(n×n\)\). In fact, using the notation from Section , these matrices are given by

({{\(l\&^\)=(({{(Subscript[p, 10]), \(p\_\(11\)\), ⋯, (Subscript[p, 1(n - 
                                  1)])}, {⋮, ⋮, ⋱, ⋮}, {(Subscript[p, n0]), \(p\_\(n1\)\), ⋯, (Subscript[p, n(
                              n - 1)])}})) ,, \(r\&^\)=(({{(Subscript[q, 10]), (Subscript[q, 11]), ⋯, (Subscript[q, 1(n - 
                                1)])}, {⋮, ⋮, ⋱, ⋮}, {(Subscript[q, n0]), (Subscript[q, n1]), ⋯, (Subscript[q, n(
                            n - 1)])}}))}}) .

\(P = ⌈w\&^\_1⌉\ \(\((l\&^w\&^\^← + 
                r\&^\(w\&^\^ -> \))\)\^\(-1\)\) \((Ll\&^\(D\&^\)\_n + 
            Rr\&^\(D\&^\)\_n)\)\)

Let us make some brief remarks on this formula. Note that \(TraditionalForm\`w\&^\_1\) denotes the first row of the fundamental matrix \(TraditionalForm\`w\&^\), which is nothing else than the corresponding fundamental system, written as a row vector. The notation \(TraditionalForm\`w\&^\^←\) and \(TraditionalForm\`\(w\&^\^ -> \)\) is used in a sense similar to what we had before, but applied to the whole matrix \(TraditionalForm\`w\&^\); it denotes the fundamental matrix evaluated at the left and right endpoint, respectively. Let us also remark that the computational complexity of determining the kernel projector P by Lemma  is not as bad as one might think at a first glance: The matrix inversion is only carried out in (ℂ^(n×n)) since both the boundary matrices \(TraditionalForm\`l\&^, r\&^\) and the evaluated fundamental matrices \(TraditionalForm\`w\&^\^←, \(w\&^\^ -> \)\) are constants. The rest consists essentially of some matrix–matrix and matrix–vector multiplications.
The boundary value problem TG = 1 and \(TraditionalForm\`B\_1G = \(… = \(B\_nG = 0\)\)\) is now reduced to the single equation GT = 1 - P, where P is the kernel projector computed from (Subscript[B, 1]), …, (Subscript[B, n]) and a fundamental system of T according to Lemma . So all we need now is some right inverse (T^◆) of T; then the desired Green's operator will be given as G = (1 - P) (T^◆). Before giving a construction for such a right inverse, let us mention that (T^◆) is also known as the fundamental solution of the differential equation Tu = f; obviously it is the Green's operator if one chooses for P the zero projector—in this sense, a fundamental solution is a Green's operator disrespecting the boundary conditions. More specifically, the concept of fundamental solution found in the literature, like [36], refers to the integral kernel γ of the operator (T^◆), just as the Green's function g is the kernel of the Green's operator G when written in kernel form; it can be extracted from (T^◆) in a completely similar manner, just as described at the end of Section . By definition, the fundamental solution fulfills T(T^◆) = 1, which implies that \(TraditionalForm\`u := \(T\^◆f = \(\([[\)\(∫\_0\%1\( γ(
                t, τ)\) \(f(τ)\)\)\(]]\)\)\)\) fulfills the differential equation Tu = f, but usually not the boundary conditions \(TraditionalForm\`B\_1u = \(… = \(B\_nu = 0\)\)\).
Now for the construction of (T^◆). If T is a differential operator with constant coefficients, it is very easy to find a right inverse. Just as with any complex polynomial, T splits into linear factors as \(TraditionalForm\`T = \((D - λ\_1)\)\(⋯(D - λ\_n)\)\). If we only know how to right–invert such first–order operators D - λ, we have \(TraditionalForm\`T\^◆ = \((D - λ\_n)\)\^◆\(⋯(D - λ\_1)\)\^◆\). But the right inverse of a first–order operator D - λ can be readily found as \(TraditionalForm\`⌈e\^\(λt\)⌉ 
        A ⌈e\^\(\(-λ\)t\)⌉\). Let us check this:

\(D ⌈e\^\(λt\)⌉ 
        A ⌈e\^\(\(-λ\)t\)⌉ = \(\((λ ⌈e\^\(λt\)⌉ + ⌈e\^\(λt\)⌉ D)\) 
          A ⌈e\^\(\(-λ\)t\)⌉ = \(λ ⌈e\^\(λt\)⌉ 
              A ⌈e\^\(\(-λ\)t\)⌉ + ⌈e\^\(λt\)⌉ 
              DA ⌈e\^\(\(-λ\)t\)⌉ = λ ⌈e\^\(λt\)⌉ 
              A ⌈e\^\(\(-λ\)t\)⌉ + 1\)\)\)

\(\((D - λ)\) ⌈e\^\(λt\)⌉ 
        A ⌈e\^\(\(-λ\)t\)⌉ = \(D ⌈e\^\(λt\)⌉ 
            A ⌈e\^\(\(-λ\)t\)⌉ - λ ⌈e\^\(λt\)⌉ 
            A ⌈e\^\(\(-λ\)t\)⌉ = \(λ ⌈e\^\(λt\)⌉ 
              A ⌈e\^\(\(-λ\)t\)⌉ + 
            1 - λ ⌈e\^\(λt\)⌉ 
              A ⌈e\^\(\(-λ\)t\)⌉ = 1\)\)\)

\(T\^◆ = ∏\+\(i = 1\)\%n⌈e\^\(λ\_it\)⌉ 
          A ⌈e\^\(\(-λ\_i\)t\)⌉\)

Given a linear differential operator with constant coefficients in the form T = 
        (Subscript[a, n])(D^n) + 
          (Subscript[a, n - 1])(D^(n - 1)) + … + 
          (Subscript[a, 1])D + (Subscript[a, 0]), one determines the roots (Subscript[λ, 1]), …, (Subscript[λ, n]) of its so–called characteristic polynomial (Subscript[a, n])(x^n) + 
          (Subscript[a, n - 1])(x^(n - 1)) + … + 
          (Subscript[a, 1])x + (Subscript[a, 0]) ∈ ℂ[
          x]; then one has of course the desired factorization \(TraditionalForm\`T = \(a\_n(
            D - λ\_1)\)\(⋯(D - λ\_n)\)\). Note that multiple roots have to repeated according to their multiplicity.
The general case of linear differential operators with variable coefficients is slightly more complicated. But it turns out that right inversion is essentially an iteration of a procedure typically called "reduction of order" in the literature; see e.g. page 84 in [37].

\(TraditionalForm\`⟨φ\_\(jk\) | j = 1, …, n ∧ k = j, …, 
      n⟩\)

({{\(Φ = ∏\+\(k = 1\)\%n\((⌈φ\_\(kk\)⌉ A)\)\), and, \(φ = \(a\_n\) \(∏\+\(k = 1\)\%n φ\_\(kk\)\)\)}}) ,

\(TraditionalForm\`\(\(TΦ = \((⌈a\_n⌉ 
              D\^n + ⌈a\_\(n - 1\)⌉ 
              D\^\(n - 1\) + … + ⌈a\_1⌉ 
              D + ⌈a\_0⌉)\)\ ⌈φ\_\(11\)⌉ 
        A ⌈φ\_\(22\)⌉ 
        A⋯⌈φ\_\(nn\)⌉ 
        A\)\(\ \)\(,\)\)\)

Let us illustrate the whole procedure of solving a BVP by the simple and yet highly instructive example mentioned at the beginning of this chapter—the heat equation (in the indicated special case). In its functional formulation, the problem is this: Given f ∈ (C^∞)[0, 1], find u ∈ (C^∞)[0, 1] such that u'' = f and u(0) = u(1) = 0. Written as an operator equation with integro–differential polynomials, we have to find \(TraditionalForm\`G ∈ 𝒜𝓃(𝔈𝔵𝔭)\) with (D^2)G = 0 and LG = RG = 
          0. The nullspace of the differential operator (D^2) is given by the linear functions \(TraditionalForm\`\(\([[\)\(αt + β\)\(]]\)\)\) for all \(TraditionalForm\`α, β ∈ ℂ\); the computation of the projector yields

P = 
      L - ⌈t⌉ 
          L + ⌈t⌉ R .

\(u \((t)\) = \(∫\_0\%1\( γ(
              t - τ)\) \(f(τ)\) dτ = ∫\_0\%t\((t - τ)\) \(f(τ)\) dτ\)\)

G=(1 - P)((((D^2)))^◆)=(L - ⌈t⌉ 
                    L + ⌈t⌉ R) (A^2)=(L - ⌈t⌉ 
                    L + ⌈t⌉ R)(⌈t⌉ A - 
                    A ⌈t⌉)

G = ⌈t⌉ 
          A ⌈A⌉ + ⌈t⌉ 
          B ⌈A⌉ - 
        A ⌈t⌉ - ⌈t⌉ 
          B ,

g(t, τ)={({{(t - 1) τ, ⇐, 0 ≤ τ ≤ 
                t ≤ 1 ,}, {(τ - 1) t, ⇐, 0 ≤ 
                t ≤ τ ≤ 1}})

The Generic Sturm–Liouville Problem

({{u'' = f}, {u(0) = u(1) = 0}})

Summing up, a Sturm–Liouville BVP has the following form: Given f ∈ (C^∞)[0, 1], find u ∈ (C^∞)[0, 1] such that:

({{u'' + a u' + b u = f}, {α u(0) + β u' (0) = 0}, {γ u(1) + δ u' (1) = 0}})

The Sturm–Liouville problem as stated above can be solved nicely by an analytic method; see e.g. page 195 in [26]. The purpose of this section is to analyze the generic Sturm–Liouville problem by the algebraic machinery of the Moore–Penrose calculus. In order to keep the computations simple, we will henceforth assume constant coefficients a, b ∈ ℂ. In this case, the whole theory we have developed so far can be used if we simply replace the scalar field ℂ used so far by the rational–function field ℂ(a, 
        b, α, β, γ, δ). It turns out that we can then solve the problem in essentially the same way as we may treat any of the "concrete" problems.
First of all, we have to repeat our restriction to well–posed problems (see the next section for getting an impression of what happens in the ill–posed case). In the current situation, this translates into the well–known coefficient condition

\[LeftBracketingBar]({{αφ(
                      0) + βφ' (0), αψ(
                      0) + βψ' (0)}, {γφ(
                      1) + δφ' (1), γψ(
                      1) + δψ' (1)}})\[RightBracketingBar]≠0 ,

Now let us introduce some abbreviations: The differential operator of the problems will be denoted by T = (D^2) + a D + b, and the two boundary operators by M = α L + β LD and N = γ R + δ RD. With \(TraditionalForm\`λ, μ ∈ ℂ\) being the two roots of the characteristic polynomial (x^2) + ax + b, the differential operator factors as T = (D - λ) (D - μ). Hence Lemma  tells us that

(T^◆) = ⌈(e^(λt))⌉ 
        A ⌈(e^((μ - λ) t))⌉ 
        A ⌈(e^(-μt))⌉

The next step in the Moore–Penrose solution process consists of computing the kernel projector. Using the notation of Lemma , we have here

({{\(l\&^\)=(({{α, β}, {0, 0}})) ,, \(r\&^\)=(({{0, 0}, {γ, δ}}))}})

\((L\(l\&^\) + R\(r\&^\))\)(({{1}, {D}}))=(({{M}, {N}})) .

φ(
        t) = (((α + βλ) (e^(μt)) - (α + βμ) (e^(λt)))/(λ - μ))

ψ(
        t) = (((γ + δλ) (e^(μ(t - 1))) - (γ + δμ) (e^(λ(t - 1))))/(λ - μ))

\(\(l\&^\)\(\(w\&^\)\^ ← \) + \(r\&^\)\(\(w\&^\)\^ -> \)\)=(({{Mφ, Mψ}, {Nφ, Nψ}}))=(({{0, m}, {n, 0}}))

m := Mψ = (((α + βμ) (γ + δλ) (e^(-μ)) - (α + βλ) (γ + δμ) (e^(-λ)))/(λ - μ)) ,

n := Nφ = (((α + βλ) (γ + δμ) 
                  (e^μ) - (α + βμ) (γ + δλ) 
                  (e^λ))/(λ - μ)) .

\(\((\(l\&^\)\(\(w\&^\)\^ ← \) + \(r\&^\)\(\(w\&^\)\^ -> \))\)\^\(-1\)\)=(((({{0, m}, {n, 0}})))^(-1))=-(1/(mn))(({{0, -m}, {-n, 0}}))=(m^(-1)) (({{0, -(e^(-λ - μ))}, {1, 0}})) ,

P=\(⌈w\&^\_1⌉\)\(\((l\&^w\&^\^← + 
                  r\&^\(w\&^\^ -> \))\)\^\(-1\)\)\((Ll\&^ + Rr\&^)\)(({{1}, {D}}))=(m^(-1)) (({{⌈φ⌉, ⌈ψ⌉}}))(({{0, -(e^(-λ - μ))}, {1, 0}}))(({{M}, {N}}))
=(m^(-1)) (({{⌈φ⌉, ⌈ψ⌉}})) (({{-(e^(-λ - μ)) N}, {M}}))=(m^(-1)) (⌈ψ⌉ 
                M - (e^(-λ - μ)) ⌈φ⌉ N)

With these preparations, we can in principle write down the desired Green's operator for the generic Sturm–Liouville problem (under the additional assumptions we have made). Since G = (1 - P) (T^◆), we have

m G = (m - ⌈ψ⌉ 
              M + (e^(-λ - μ)) ⌈φ⌉ 
              N) ⌈(e^(λt))⌉ 
        A ⌈(e^((μ - λ) t))⌉ 
        A ⌈(e^(-μt))⌉ .

In this form, the Green's operator cannot yet be realized by a Green's function. In order to obtain a canonical form of G, one just has to employ the reduction machinery of the Green's system (of course including the definitions of M, N and m as well as φ and ψ). Let us only quote the output of this computation here, which is a rather lengthy expression with eight summands:

m G = (α + βμ) (γ + δμ) (((λ - μ))^(-2)) (e^(-λ)) ⌈(e^(λt))⌉ 
          A ⌈(e^(-μt))⌉ + (α + βμ) (γ + δμ) (((λ - μ))^(-2)) (e^(-λ)) ⌈(e^(λt))⌉ 
          B ⌈(e^(-μt))⌉ - 
(α + βλ) (γ + δμ) (((λ - μ))^(-2)) (e^(-λ)) ⌈(e^(μt))⌉ 
          B ⌈(e^(-μt))⌉ - (α + βλ) (γ + δμ) (((λ - μ))^(-2)) (e^(-λ)) ⌈(e^(λt))⌉ 
          A ⌈(e^(-λt))⌉ + 
(α + βλ) (γ + δλ) (((λ - μ))^(-2)) (e^(-μ)) ⌈(e^(μt))⌉ 
          A ⌈(e^(-λt))⌉ + (α + βλ) (γ + δλ) (((λ - μ))^(-2)) (e^(-μ)) ⌈(e^(μt))⌉ B ⌈(e^(-λt))⌉ - 
(α + βμ) (γ + δλ) (((λ - μ))^(-2)) (e^(-μ)) ⌈(e^(μt))⌉ 
          A ⌈(e^(-μt))⌉ - (α + βμ) (γ + δλ) (((λ - μ))^(-2)) (e^(-μ)) ⌈(e^(λt))⌉ 
          B ⌈(e^(-λt))⌉

One can immediately extract the Green's function from this integro–differential in the usual manner. For the sake of a more economical representation (and also for an easy comparison with the literature), it is benefitial to factor this expression:

(((λ - μ))^2) m G=((γ + δλ) (e^(-μ)) ⌈(e^(μt))⌉ - (γ + δμ) (e^(-λ)) ⌈(e^(λt))⌉)A ((α + βλ) ⌈(e^(-λt))⌉ - (α + βμ) ⌈(e^(-μt))⌉)+
((α + βλ) ⌈(e^(μt))⌉ - (α + βμ) ⌈(e^(λt))⌉)B ((γ + δλ) (e^(-μ)) ⌈(e^(-λt))⌉ - (γ + δμ) (e^(-λ)) ⌈(e^(-μt))⌉)

(((λ - μ))^2) m G=((γ + δλ) ⌈(e^(μ(t - 1)))⌉ - (γ + δμ) ⌈(e^(λ(t - 1)))⌉)A ((α + βλ) ⌈(e^(-λt))⌉ - (α + βμ) ⌈(e^(-μt))⌉)+
((α + βλ) ⌈(e^(μt))⌉ - (α + βμ) ⌈(e^(λt))⌉)B ((γ + δλ) ⌈(e^(-(λ + μ)t))⌉ - (γ + δμ) ⌈(e^(-(λ + μ)t))⌉) .

(((λ - μ))^2) m G=((λ - μ) ⌈ψ⌉)A ((λ - μ) ⌈(e^(-(λ + μ) t)) φ⌉)+((λ - μ) ⌈φ⌉)B ((λ - μ) ⌈(e^(-(λ + μ) t)) ψ⌉)

G = ⌈ψ⌉ 
          A ⌈(m^(-1))(e^(-(λ + μ) 
                    t)) φ⌉ + ⌈φ⌉ 
          B ⌈(m^(-1))(e^(-(λ + μ) t)) ψ⌉ .

G = ⌈ψ⌉ 
              A ⌈(W^(-1))φ⌉ + ⌈φ⌉ 
              B ⌈(W^(-1))ψ]⌉

Translating this into a Green's function, we obtain

g(t, τ)={({{ψ(
                  t) φ(τ)((W(τ))^(-1)), ⇐, 0 ≤ τ ≤ 
                t ≤ 1 ,}, {φ(
                  t) ψ(τ)((W(τ))^(-1)), ⇐, 0 ≤ 
                t ≤ τ ≤ 1 ,}})

Outlook: Ill–Posed Problems

In our current context of two–point BVPs, we need not be afraid of condition 3 in Hadamard's definition: Since the solution u = Gf of a BVP like Tu = f, \(TraditionalForm\`B\_1u = \(… = \(B\_nu = 0\)\)\) is realized by an integral operator G, the input data f is smoothed out even more. As to conditions 1 and 2, however, there may be some problems. As a matter of fact, we have always assumed up to now that the boundary conditions are set up such that there is a unique solution, thus stipulating these two conditions from the outset. But what if they are not fulfilled? Here the concept of the Moore–Penrose inverse comes to our aid again: We project the forcing function f onto the image of T, such that the BVP becomes solvable. Still, this solution may not be unique, so we project again onto a normalized solution. Usually one chooses orthogonal projectors (in the sense of (L^2)) in both cases. It turns out that the corresponding (generalized) solution operator can still be expressed as an integral operator whose kernel is sometimes called the modified Green's function; see p. 210 of [26]. In the sequel, we will just speak of the Green's operator and the Green's function.
The treatment of such ill–posed problems by the Moore–Penrose calculus is still in its infancy. In order to get an impression of what happens, let us look at the following small but instructive example:

({{-u'' = f}, {u' (0) = u' (1) = 0}})

So we will have to restore both existence and uniqueness of the solution in our small example. In order to enforce existence, we will project an arbitrary forcing function f ∈ (C^∞)[0, 1] orthogonally onto the subspace of those forcing functions that fulfill the solvability condition. Writing F ≡ A + B, the obvious projector for doing this is 1 - F. Now for realizing uniqueness, one typically selects the solution with minimal (L^2) norm (which amounts to another orthogonal projector). Since the kernel of the BVP consists of the constant functions, the norm of u can always be made zero. Hence we add the integral condition (∫_0^1) 
              u(τ) dτ = 0 to the two given boundary conditions. At this point it is crucial that the Green's algebra can incorporate integral (global) conditions as well as boundary (local) conditions. We can summarize the new problem of finding the "best–approximate solution" as follows:

({{-u'' = (1 - F) f}, {u' (0) = u' (1) = (∫_0^1) 
                      u(τ) \[DifferentialD]τ = 0}})

Due to our modifications, this "boundary value problem" is a regular one; we will therefore refer to it as the "regularized problem". But can we apply the solution method described above? Obviously we cannot use the old method in a straight–forward way: First of all, the Green's operator is now a right inverse of 1 - F rather than 1; and second the projector onto the kernel of -(D^2) gives us only two degrees of freedom, so we cannot satisfy three conditions. Speaking of the limitations of the method previously described, let us pause for a moment and ask ourselves what happens when we apply it to the BVP of this section, i.e. to solve -(D^2)G = 1, \(TraditionalForm\`LDG = \(RDG = 0\)\) directly. Obviously, it is not difficult to come up with a right inverse for -(D^2); up to normalzation, it is simply -(A^2). Hence the solution formula would yield G = (P - 1) (A^2), but the problem is hidden in the projector P. For applying Lemma , we compute:

({{\(w\&^\)=(({{1, t}, {0, 1}})), ⟹, \(\(w\&^\)\^ ← \)=(({{1, 0}, {0, 1}})),, \(\(w\&^\)\^ -> \)=(({{1, 1}, {0, 1}}))}})

({{\(l\&^\)=(({{0, 1}, {0, 0}})),, \(r\&^\)=(({{0, 0}, {0, 1}}))}})

\(\(l\&^\)\(\(w\&^\)\^ ← \) + \(r\&^\)\(\(w\&^\)\^ -> \)\)=(({{0, 1}, {0, 0}}))+(({{0, 0}, {0, 1}}))=(({{0, 1}, {0, 1}})) ,

The key to solving the regularized BVP is to transform it to an equivalent auxiliary problem that is accessible to the methods described earlier. The idea is quite simple: Taking one more derivative of the differential equation leads to the following BVP in "classical" form (except for the global third condition, which is unproblematic as we will see):

({{-u''' = f'}, {u' (0) = u' (1) = (∫_0^1) 
                      u(τ) \[DifferentialD]τ = 0}})

Cw (t)=\(∫\_0\%1\(∫\_τ\%t\( 
                u(τ\&_)\) \[DifferentialD]τ\&_ \[DifferentialD]τ\)\)

\(\(C\&^\)w\ \((t)\)\)=\(∫\_0\%t\(∫\_0\%\(τ\&_\)\(u(τ\&_)\) \[DifferentialD]τ \[DifferentialD]τ\&_\)\)-\(∫\_t\%1\(∫\_\(τ\&_\)\%1\( 
                u(τ\&_)\) \[DifferentialD]τ \[DifferentialD]τ\&_\)\)

\(\(C\&^\)w\ \((t)\)\)=(∫_0^t) τ u(τ) \[DifferentialD]τ-(∫_t^1)(1 - τ) u(τ) \[DifferentialD]τ=(∫_0^t) τ u(τ) \[DifferentialD]τ-(∫_0^1)(1 - τ) u(τ) \[DifferentialD]τ+(∫_0^t)(1 - τ) u(τ) \[DifferentialD]τ=(∫_0^t) 
                    u(τ) \[DifferentialD]τ-(∫_0^1)(1 - τ) u(τ) \[DifferentialD]τ .

Whereas the standard antiderivative A is characterized by

(∫_0^t) 
            u' (τ) \[DifferentialD]τ = 
      u (t) - u (0) ,

\(\(\(∫\_0\%1\(∫\_τ\%t 
              u' \((τ\&_)\) \[DifferentialD]τ\&_\ \[DifferentialD]τ\) = 
      u \((t)\) - ∫\_0\%1\( 
              u \((τ)\)\) \[DifferentialD]τ\)\(\ \)\(,\)\)\)

-u'' + (∫_0^1) 
              u'' (τ) \[DifferentialD]τ = f - (∫_0^1) 
                f (τ) \[DifferentialD]τ = (1 - F) f ,

We can now proceed to solve the auxiliary problem with the standard method described earlier. The kernel of -(D^3) is given by the quadratic polynomials, so it has dimension 3. Hence we can choose a projector P onto it such that its complement consists exactly of those functions that fulfill the three side conditions of the auxiliary problem (analogous to what Lemma  does if we have only local conditions). Using the ansatz Pu (t) = α[u] 
            (t^2) + β[u] t + γ[u], we obtain

P = 
      F - (1/3) LD - (1/6) 
          RD + ⌈t⌉ 
          LD + (1/2) ⌈(t^2)⌉ (RD - LD)

The corresponding Green's operator \(TraditionalForm\`G\&~\) fulfills \(TraditionalForm\`\(-G\&~\)D\^3 = 1 - P\), but note that \(TraditionalForm\`G\&~\) must be applied to f' rather than f. Hence the actual Green's operator G of the regularized problem (which maps f to u) is given by \(TraditionalForm\`G = G\&~D\), and we have -G(D^2) = 
        1 - P or G = (P - 1) (A^2). (This result is just as if we had started from a well–posed second–order BVP with projector P; but as we have seen, the above projector P is really coming from a third–order problem.) Normalizing this result with respect to the Green's system, we finally obtain

G = (1/2) A ⌈(t^2)⌉ + (1/2) 
          B ⌈(t^2)⌉ + (1/2) ⌈(t^2)⌉ A + (1/2) ⌈(t^2)⌉ B + (1/3) 
          A + (1/3) B - ⌈t⌉ A - 
        B ⌈t⌉ .

As usual, we may optionally extract the Green's function from this as

g(t, τ)={({{(((t^2) + (τ^2))/2) - t + (1/3), ⇐, 0 ≤ τ ≤ 
                t ≤ 1 ,}, {(((t^2) + (τ^2))/2) - τ + (1/3) - , ⇐, 0 ≤ 
                t ≤ τ ≤ 1 ,}})