On Products of Distributions in Colombeau Algebra

Abstract

Results on products of distributions and are derived. They are obtained in Colombeau differential algebra of generalized functions that contains the space of Schwartz distributions as a subspace. Products of this form are useful in quantum renormalization theory in Physics.

1. Introduction

In quantum physics one find the need to evaluate , when calculating the transition rates of certain particle interactions see [1]. The problem of defining products of distributions is also closely connected with the problem of renormalization in quantum field theory. Due to the large use of distributions in the natural sciences and other mathematical fields, the problem of the product of distributions [2] is an objective of many research studies. Starting with the historically first construction of distributional multiplication by König [3] and the sequential approach developed in [4] by Antosik et al., there have been numerous attempts to define products of distributions, see [5–7], or rather to enlarge the number of existing products.

In order to define a general product of distributions the mathematicians were seeking an algebra containing , having the constant function as a unit element and such that there is a derivation which is linear, which generalizes the derivation in the sense of distributions and such that Leibnitz's rule for derivation of a product holds. With his famous impossibility result Schwartz [8] showed that the new product in cannot coincide with the usual product of continuous functions.

The actual construction of such algebras enjoying these optimal properties is due to Colombeau [9]. His theory of algebras of generalized functions gave a rigorous mathematical sense to the “heuristic” computations that were used in quantum field theory. In his theory Colombeau deals with generalized functions which are defined on any open subset of and are more general than the distributions on . From the viewpoint of differentiation these new generalized functions have the same properties as distributions. It is from the viewpoint of multiplication and nonlinear operations that these new generalized functions have properties completely different from those of distributions: any finite product of generalized functions is still generalized function and, more generally, the algebras of these generalized functions are closed with respect to many nonlinear operations, so that any finite product of distributions is a generalized function and not a distribution in general. However, these new generalized functions are very much related to the distributions in the sense that their definition may be considered as a natural evolution of Schwartz's definition of distribution.

In a few words, the differential Colombeau algebra as a powerful tool for solving linear and nonlinear problems includes singularities (see [10]) and has almost the optimal properties while the problem of multiplication of Schwartz’s distributions is concerned: it is an associative differential algebra of generalized functions and contains the algebra of smooth functions as a subalgebra; the distribution space is linearly embedded in it as a subspace and the multiplication is compatible with the operations of differentiation and products with -differentiable functions. The notion “association” in is a faithful generalization of the equality of distributions and enables obtaining results in terms of distributions again.

Following this approach, in this paper we evaluate some products of distributions, as embedded in Colombeau algebra, in terms of associated distributions. Similar results can be found in [11]. The results obtained in this way can be reformulated as regularized products in the classical distribution theory.

2. Colombeau Algebra

In this section we will introduce basic notations and definitions from Colombeau theory.

Let be the set of nonnegative integers, that is, . For we denote where is the space of all functions with compact support. The elements of the set are called test functions. It is obvious that . Also, for all For and it is denoted, and .

Wanting to obtain an algebra containing the space of distributions, whose elements could be multiplied and differentiated as well as functions, Colombeau started with , the algebra of functions that are infinitely differentiable with respect to the second variable, . The embedding of distributions into such an algebra must be done in a way that the embedding of functions will be identity. Let and be functions. Taking the sequence , which converges to in , as a representative of , we obtain an embedding of into . So, if we consider and as distributions, we look at the sequences and . The product of and as distributions does not always coincide with their classical product considered as a distribution; that is, The idea therefore is to find an ideal such that this difference will vanish in the resulting quotient. In order to determine it is obviously enough to find an ideal containing the differences .

Expanding the last term in a Taylor’s series and having in mind properties of as an element of we can see that it will vanish faster than any power of , uniformly on compact sets, in all derivatives. The set of these differences will not be an ideal in but in a set of a sequences whose derivatives are bounded uniformly on compact sets by negative power of . These sequences are called “moderate” sequences and the set containing them is denoted by .

Finally, the generalized functions of Colombeau are elements of the quotient algebra where, as explained before, is the subalgebra of “moderate” functions such that for each compact subset of and any there is a such that, for each , there are , and it holds for and is an ideal of consisting of all functions such that for each compact subset of and any there is a such that for every and each there are and it holds: for . Elements of are also known as “null” functions or “negligible” functions.

The Colombeau algebra contains the distributions on canonically embedded as a -vector subspace by the map: where denotes the convolution product of two distributions and is given by

According to the above, we can also write: , where denotes the integral .

An element (a generalized function of Colombeau) is actually an equivalence class of an element which is called representative of . Multiplication and differentiation of generalized functions are performed on arbitrary representatives of the respective generalized functions.

With the next two definitions, we introduce the notion of “association”.

Definition 1. Generalized functions , are said to be associated and denoted by , if for some representatives and and arbitrary there is a such that for any

Definition 2. A generalized function is said to admit some as “associated distribution”, denoted by , if for some representative of and any there is a such that for any These definitions are independent of the representatives chosen and the distribution associated, if it exists, is unique. The association is a faithful generalization of the equality of distributions.

Multiplying two distributions in as a result it is in general obtained a generalized function which may not always be associated with the third distribution. By Colombeau product of distributions it is meant the product of their embedding in whenever the result admits an associated distribution. If the regularized model product of two distributions exists, then their Colombeau product also exists and it is the same with the first one. The relation is asymmetric; the distribution stands on the r.h.s.; the relation is an equivalent relation in so it is symmetric in and it can also be written as . As a final conclusion of this introduction we have a fact that we operate with the elements of exactly the same as with the -functions, because we actually operate with their representatives which are -functions.

3. Results on Some Products of Distributions

Recently, Li in [12] considered several results concerning commutative product of distributions and proved that In [13] it was proved that for any the product of the generalized functions and in holds: Also in [14] the product of the generalized functions and in was considered and it was proved that In this paper we will make a generalization of (11), first for the one-dimensional case. Products of the form are useful in quantum renormalization theory in Physics.

Theorem 3. The product of the generalized functions and for and in admits associated distributions and it holds

Proof. For given we suppose that , without loss of generality. By definition we have . Then using the embedding rule and the substitution we have the representatives of the distribution in Colombeau algebra:
Differentiating in , we get the representative of :
Then, for any we have using the substitution . By the Taylor theorem we have for . Using relation (17) in (16) and changing the order of integration we have where and .
Next using integration by part we have The first term above is zero, and we have where . The integral is nonzero only if , and its value is . Thus the only nonzero term in the sum above is obtained when ; that is, .
Consider Therefor, passing to the limit, as , we obtain (13) proving the theorem.

Next, to extend our main result in we need the following lemma proved in [15].

Lemma 4. Let and be distributions in such that and with each and in and suppose that their embedding in satisfy , for . Then , where is in .

Theorem 5. The product of the generalized functions and for and in admits associated distributions and it holds

Proof. In this case, due to the tensor product structure of the distributions that are considered, we can apply previous lemma and we have which completes the proof of the theorem.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

This work is supported by the research project “Functional Spaces, Topological and Statistical Aspects and Their Application in Electrical Engineering” and financed by State University “Goce Delce”, Stip, Macedonia.