An Embedding of Schwartz Distributions in the Algebra of Asymptotic Functions

We present a solution of the problem of multiplication of Schwartz distributions by embedding the space of distributions into a differential algebra of generalized functions, called in the paper ``asymptotic function'', similar to but different from J. F. Colombeau's algebras of new generalized functions.


Introduction
The main purpose of this paper is to prove the existence of an embedding Σ D,Ω of the space of Schwartz distributions D ′ (Ω) into the algebra of asymptotic functions ρ E(Ω) which preserves all linear operations in D ′ (Ω).Thus, we offer a solution of the problem of multiplication of Schwartz distributions since the multiplication within D ′ (Ω) is impossible (L.Schwartz [1]).
The algebra ρ E(Ω) is defined in the paper as a factor space of nonstandard smooth functions.The field of the scalars ρ C of the algebra ρ E(Ω) coincides with the complex counterpart of the Robinson asymptotic numbers -known also as Robinson's field with valuation (Robinson's (2]) and Lightstone and Robinson [3]).The embedding Σ D,Ω is constructed in the form Σ D,Ω = Q Ω •D⋆Π• * , where (in backward order): * is the extension mapping (in the sense of nonstandard analysis), • is the Schwartz multiplication in D ′ (Ω) (more precisely, its nonstandard extension), ⋆ is the convolution operator (more precisely, its nonstandard extension), • denotes "composition", Q Ω is the quotient mapping (in the definition of the algebra of asymptotic functions) and D and Π Ω are fixed nonstandard internal functions with special properties whose existence is proved in this paper.
Our interest in the algebra ρ E(Ω) and the embedding D ′ (Ω) ⊂ ρ E(Ω), is due to their role in the problem of multiplication of Schwartz distributions, the nonlinear theory of generalized functions and its applications to partial differential equations (Oberguggenberger [4]), (Todorov [5] and [6]).In particular, there is a strong similarity between the algebra of asymptotic functions ρ E(Ω) and its generalized scalars ρ C, discussed in this paper, and the algebra of generalized functions G(Ω) and their generalized scalars C, introduced by J. F. Colombeau in the framework of standard analysis (Colombeau [7], pp.63, 138 and Colombeau [8], §8.3, pp.161-166).We should mention that the involvement of nonstandard analysis has resulted in some improvements of the corresponding standard counterparts; one of them is that ρ C is an algebraically closed field while its standard counterpart C in J. F. Colombeau's theory is a ring with zero divisors.
This paper is a generalization of some results in [9] and [10] (by the authors of this paper, respectively) where only the embedding of the tempered distributions S ′ (R d ) in ρ E(R d ) has been established.The embedding of all distributions D ′ (Ω), discussed in this paper, presents an essentially different situation.We should mention that the algebra ρ E(R d ) was recently studied in (Hoskins and Sousa Pinto [11]).
Here Ω denotes an open set of R d (d is a natural number), E(Ω) = C ∞ (Ω) and D(Ω) = C ∞ 0 (Ω) denote the usual classes of C ∞ -functions on Ω and C ∞ -functions with compact support in Ω and D ′ (Ω), and E ′ (Ω) denote the classes of Schwartz distributions on Ω and Schwartz distributions with compact support in Ω, respectively.As usual, N, R, R + , and C will be the systems of the natural, real, positive real and complex numbers, respectively, and we use also the notation N 0 = {0} ∪ N.For the partial derivatives we write For a general reference to distribution theory we refer to Bremermann [12] and Vladimirov [13].
Our framework is a nonstandard model of the complex numbers C, with degree of saturation larger than card(N).We denote by * R, * R + , * C, * E(Ω) and * D(Ω) the nonstandard extensions of R, R + , C, E(Ω) and D(Ω), respectively.If X is a set of complex numbers or a set of (standard) functions, then * X will be its nonstandard extension and if f : X → Y is a (standard) mapping, then * f : * X → ⋆ Y will be its nonstandard extension.For integration in * R d we use the * -Lebesgue integral.We shall often use the same notation, x , for the Euclidean norm in R d and its nonstandard extension in ⋆ R d .For a short introduction to nonstandard analysis we refer to the Appendix in Todorov [6].For a more detailed exposition we recommend Lindstrom [14], where the reader will find many references to the subject.

Test Functions and Their Moments
In this section we study some properties of the test functions in D(R d ) (in a standard setting) which we shall use subsequently.
In addition to the above we have the following result: More precisely, for any positive real δ there exists ϕ in A k such that In addition, ϕ can be chosen symmetric.
Proof.We consider the one dimensional case d = 1 first.Start with some fixed positive (real valued) ψ in D(R) such that ψ(x) = 0 for |x| ≥ 1 and R ψ(x)dx = 1 (ψ can be also chosen symmetric if needed).We shall look for ϕ in the form: for i = 0, 1, . . ., k, we derive the system for linear equations for c j : The system is certainly satisfied, if which can be written in the matrix form V k+1 (ε) C = B, where V k+1 (ε) is Vandermonde (k + 1) × (k + 1) matrix, C is the column of the unknowns c j , and B is the column whose top entry is 1 and all others are 0.For the determinant we have det V k+1 ( ε) = 0 for ε = 1, therefore, the system has a unique solution (c 0 , c 1 , c 2 , . . ., c k ).Our next goal is to show that this solution is of the form: (2.3) , where P j and P are polynomials and (2.4) for 0 ≤ j ≤ k, and The coefficients c 0 , c 1 , c 2 , . . ., c k will be found by Cramer's rule.The formula for Vandermonde determinants gives for some polynomial P , where To calculate the numerator in (2.3), we have to replace the jth column of the matrix by the column B (whose top entry is 1 and all others are 0) and calculate the resulting determinant D j .Consider first the case 1 ≤ j ≤ k − 1.By developing with respect to the jth column, we get We factor out ε 2 , ε 4 , . . ., ε 2(j−1) , ε 2(j+1) , . . ., ε 2k and obtain: The latter is a Vandermonde determinant again, and we have Hence, factoring out ε (i−1)(k−i) in the ith row above, we get D j = ±ε α j (1 + εP j (ε)) for some polynomials P j (ε) and which coincides with the desired result (2.4) for α j , in the case 1 ≤ j ≤ k − 1.For the extreme cases j = 0 and j = k, we obtain which both can be incorporated in the formula (2.4) for α j .Finally, Cramer's rule gives the expression (2.3) for c j .Now, taking into account that ψ ≥ 0, by assumption, and the fact that and this latter expression can be made smaller than 1 + δ for sufficiently small ε if a) ) is obvious, as for a), we have: To generalize the result for arbitrary dimension d, it suffices to consider a product of functions of one real variable.The proof is complete.

Nonstandard Delta Functions
We prove the existence of a nonstandard function D in * D(R d ) with special properties.The proof is based on the result of Lemma 2.2 and the Saturation Principle (Todorov [6], p. 687).We also consider a type of nonstandard cut-off-functions which have close counterparts in standard analysis.The applications of these functions are left for the next sections.
where ≈ is the infinitesimal relation in * C. We shall call this type of function nonstandard ρ-mollifiers Proof.For any k ∈ N, we define the set of test functions: and the internal subsets of * D(R d ): Obviously, we have On the other hand, we have is a real (standard) number and, hence, Theorem 3.3 (Existence).For any positive infinitesimal ρ in * R there exists a ρ-delta function.
Proof.Let θ be a nonstandard ρ-mollifier of the type described in Lemma 3.1.Then the nonstandard function D in * D(R d ), defined by Remark.The existence of nonstandard functions D in * D(R d ) with the above properties is in sharp contrast with the situation in standard analysis where there is no D in D(R d ) which satisfies both (ii) and (iii).Indeed, if we assume that For other classes of nonstandard delta functions we refer to (Robinson [15], p. 133) and to (Todorov [16]).
Our next task is to show the existence of an internal cut-off function.

Notations.
Let Ω be an open set of R d .
1) For any ε ∈ R + , we define where x is the Euclidean norm in R d , ∂Ω is the boundary of Ω and d(x, ∂Ω) is the Euclidean distance between x and ∂Ω.We also denote: 2) We shall use the same notation, ⋆, for the convolution operator ⋆ : [13]) and its nonstandard extension well as for the convolution operator ⋆ : , defined for all sufficiently small ε ∈ R + , and for its nonstandard extension: 3) Let τ be the usual Euclidean topology on R d .We denote by Ω the set of the nearstandard points in * Ω, i.e.
where µ(x), x ∈ R d , is the system of monads of the topological space (R d , τ ) (Todorov [6], p. 687).Recall that if ξ ∈ * Ω, then ξ ∈ Ω if and only if ξ is a finite point whose standard part belongs to Ω.

The Algebra of Asymptotic Functions
We define and study the algebra ρ E(Ω) of asymptotic functions on an open set Ω of R d .The construction of the algebra ρ E(Ω), presented here, is a generalization and a refinement of the constructions in [9] and [10] (by the authors of this paper, respectively), where the algebra ρ E(R d ) was introduced by somewhat different but equivalent definitions.On the other hand, the algebra of asymptotic functions ρ E(Ω) is somewhat similar to but different from the Colombeau algebras of new generalized functions (Colombeau [7], [8]).This essential difference between ρ E(Ω) and Colombeau's algebras of generalized functions is the properties of the generalized scalars: the scalars of the algebra ρ E(Ω) constitutes an algebraically closed field (as any scalars should do) while the scalars of Colombeau's algebras are rings with zero divisors (Colombeau [8], §2.1).This improvement compared with J. F. Colombeau's theory is due to the involvement of the nonstandard analysis.
Let Ω be an open set of R d and ρ ∈ * R be a positive infinitesimal.We shall keep Ω and ρ fixed in what follows.Following (Robinson [2]), we define: Definition 4.1 (Robinson's Asymptotic Numbers).The field of the complex Robinson ρ-asymptotic numbers is defined as the factor space ρ C = C M /C 0 , where (M stands for moderate).We define the embedding C ⊂ ρ C by c → q(c), where q : C M → ρ C is the quotient mapping.The field of the real asymptotic numbers is defined by It is easy to check that C 0 is a maximal ideal in C M and hence ρ C is a field.Also ρ R is a real closed totally ordered nonarchimedean field (since * R is a real closed totally ordered field) containing R as a totally ordered subfield.Thus, it follows that ρ C = ρ R(i) is an algebraically closed field, where i = √ −1.The algebra of asymptotic functions is, in a sense, a C ∞ -counterpart of A. Robinson's asymptotic numbers ρ C: Definition 4.2 (Asymptotic Functions on Ω). (i) We define the class ρ E(Ω) of the ρ-asymptotic functions on Ω (or simply, asymptotic functions on Ω if no confusion could arise) as the factor space ρ E(Ω) = E M (Ω)/E 0 (Ω), where 0 and all ξ ∈ Ω}, and Ω is the set of the nearstandard points of * Ω (3.2).The functions in E M (Ω) are called ρ-moderate (or, simply, moderate) and those in E 0 (Ω) are called ρ-null functions (or, simply, null functions).
(ii) The pairing between ρ E(Ω) and D(Ω) with values in ρ C, is defined by where q : C M → ρ C and Q Ω : E M (Ω) → ρ E(Ω) are the corresponding quotient mappings, ϕ is in D(Ω) and * ϕ is its nonstandard extension of ϕ.
is a ring and C 0 is an ideal in C M and, on the other hand, both E M (Ω) and E 0 (Ω) are closed under differentiation, by definition.Hence, the factor space ρ E(Ω) is also a differential ring.It is clear that, E M (Ω) is a module over the ring C M , and, in addition, the annihilator {c ∈ C M : (∀f ∈ E M (Ω))(cf ∈ E 0 (Ω))} of C M coincides with the ideal C 0 .Thus, ρ E(Ω) becomes an algebra over the field of the complex asymptotic numbers ρ C. (ii . By the definition of E 0 (Ω) (applied for α = 0 and n = 1), it follows f = 0 since * f is an extension of f and ρ is an infinitesimal.Thus, the mapping f → σ Ω (f ) is injective.It preserves the algebraic operations since the mapping f → * f preserves them.The preserving of the pairing follows immediately from the fact that ⋆ Ω * f (x) dx = Ω f (x) dx, by the Transfer Principle (Todorov [6], p. 686).The proof is complete.

Embedding of Schwartz Distributions
Let Ω be (as before) an open set of R d .Recall that the Schwartz embedding L Ω : L loc (Ω) → D ′ (Ω) from L loc (Ω) into D ′ (Ω) is defined by the formula: Here L loc (Ω) denotes, as usual, the space of the locally (Lebesgue) integrable complex valued functions on Ω (Vladimirov [13]).The Schwartz embedding L Ω preserves the addition and multiplication by a complex number, hence, the space L loc (Ω) can be considered as a linear subspace of D ′ (Ω).In addition, the restriction L Ω |E(Ω) of L Ω on E(Ω) (often denoted also by L Ω ) preserves the partial differentiation of any order and in this sense E(Ω) is a differential linear subspace of D ′ (Ω).In short, we have the chain of linear embeddings: The purpose of this section is to show that the algebra of asymptotic functions ρ E(Ω) contains an isomorphic copy of the space of Schwartz distributions D ′ (Ω) and, hence, to offer a solution of the Problem of Multiplication of Schwartz Distributions.This result is a generalization of some results in [9] and [10] (by the authors of this paper, respectively) where only the embedding of the tempered distributions S ′ (R d ) in ρ E(R d ) has been established.The embedding of all distributions D ′ (Ω) discussed here, presents an essentially different situation.
Remark 5.8 (Multiplication of Distributions).As a consequence of the above result, the Schwartz distributions in D ′ (Ω) can be multiplied within the associative and commutative differential algebra ρ E(Ω) (something impossible in D ′ (Ω) itself).By the property (iii) above, the multiplication in ρ E(Ω) coincides on E(Ω) with the usual (pointwise) multiplication in E(Ω).Thus, the class ρ E(Ω), endowed with an embedding Σ D,Ω , presents a solution of the problem of multiplication of Schwartz distributions which, in a sense, is optimal, in view of the Schwartz impossibility results (Schwartz [1]) (for a discussion we refer also to (Colombeau [7], §2.4) and (Oberguggenberg [18], §2).We should mention that the existence of an embedding of D ′ (R d ) into ρ E(R d ) can be proved also by sheaf-theoretical arguments as indicated in (Oberguggenberger [18], §23).Remark 5.9 (Nonstandard Asymptotic Analysis).We sometimes refer to the area connected directly or indirectly with the fields ρ R as Nonstandard Asymptotic Analysis.The fields ρ R were introduced in Robinson [2] and are sometimes known as "Robinson's nonarchimedean valuation fields".The terminology "Robinson's asymptotic numbers", chosen in this paper, is due to the role of ρ R for the asymptotic expansions of classical functions (Lightstone and Robinson [3]) and also to stress the fact that in our approach ρ C plays the role of the scalars of the algebra ρ E(Ω).Linear spaces over the field ρ R has been studied in Luxemburg [19] in order to establish a connection between nonstandard and nonarchimedean analysis.More recently ρ R has been used in Pestov [20] for studying Banach spaces.The field ρ R has been exploited by Li Bang-He [21] for multiplication of Schwartz distributions.

Lemma 3 . 1 (
Nonstandard Mollifiers).For any positive infinitesimal ρ in * R there exists a nonstandard function θ in ⋆ D(R d ) with values in * R, which is symmetric and which satisfies the following properties: for all n = 1, 2, . . ., where D denotes the Fourier transform of D. It follows D = D(0) = c for some constant c, since D is an entire function on C d , by the Paley-Wiener Theorem (Bremermann [12], Theorem 8.28, p. 97).On the other hand, D ∈ D(R d ) ⊂ S(R d ) implies D|R d ∈ S(R d ) since S(R d ) is closed under Fourier transform.Thus, it follows c = 0, i.e.D = 0 which implies D = 0 contradicting (ii).

Theorem 4 . 4 (
Differential Algebra).(i) The class of asymptotic functions ρ E(Ω) is a differential algebra over the field of the complex asymptotic numbers ρ C.