International Journal of Mathematics and Mathematical Sciences

Volume 2013 (2013), Article ID 918905, 5 pages

http://dx.doi.org/10.1155/2013/918905

Research Article

Further Results on Colombeau Product of Distributions

Biljana Jolevska-Tuneska¹ and Tatjana Atanasova-Pacemska²

¹Faculty of Electrical Engineering and Informational Technologies, Saints Cyril and Methodius University of Skopje, Karpos II bb, 1000 Skopje, Macedonia

²Faculty of Computer Sciences, Goce Delčev University of Štip, Krste Misirkov, Br 10A, 2000 Štip, Macedonia

Received 10 August 2012; Revised 28 November 2012; Accepted 10 December 2012

Academic Editor: Shyam Kalla

Copyright © 2013 Biljana Jolevska-Tuneska and Tatjana Atanasova-Pacemska. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract.

Results on Colombeau product 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 and has a notion of “association" that is a faithful generalization of the weak equality in .

1. Introduction

In quantum physics one finds 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.

Having a look at the theory of distributions [8, 9] we realize that there are two complementary points of view.

(1)A distribution is a continuous functional on the space of space functions (compactly supported smooth functions). Here we have a linear action

of on a test function .(2)Letting be a sequence of smooth functions converging to the Dirac measure, a family of regularization can be produced by convolution,

which converges weakly to the original distribution . Identifying two sequences and if they have the same limit, we obtain a sequential representation of the space of distributions. Other authors use the equivalence classes of nets of regularization. The delta-net is defined by

When we work with regularization the nonlinear structure is lost by identifying sequences (nets) with the same limit. So we have to get nonlinear theory of generalized functions that will work with regularization, but identify less. The actual construction of such algebras enjoying these optimal properties is due to Colombeau [10]. His theory of algebras of generalized functions offers the possibility of applying large classes of nonlinear operations to distributional objects. Some of them are used on solving differential equations with nonstandard coefficients [11]. The “association process” in Colombeau algebra providing a faithful generalization of the equality of distributions in enables us to obtain results in terms of distributions, the so-called Colombeau products. In fact, we evaluate particular product of distributions with coinciding singularities, as embedded in Colombeau algebra, in terms of associated distributions; see [12–14]. Therefore, the results obtained can be reformulated as regularized model products in the classical distribution theory.

In 1966, Mikusinski published his famous result [15]:

where neither of the products on the left-hand side here exists, but their difference still has a correct meaning in distribution space . Formula including balanced products of distributions with coinciding singularities can be found in mathematical and physical literature. In this paper results on product of distributions and are derived.

2. Colombeau Algebra

The basic idea underlying Colombeau’s theory in its simplest form is that of embedding the space of distributions into a factor algebra of , with regularization by convolution with a fixed “mollifier” .

The space of test function is

Functional act on test functions and points is

where is required to belong to with respect to the second variable . Now let   for  . The sequence converges to in . Taking this sequence as a representative of we obtain an embedding of into the algebra . However, embedding into this algebra via convolution as above will not yield a subalgebra since of course in general. The idea, therefore, is to factor out an ideal such that this difference vanishes in the resulting quotient. In order to construct it is obviously sufficient to find an ideal containing all differences . Taylor expansion of shows that this term will vanish faster than any power of , uniformly on compact sets, in all derivatives. We use the following space:

moderate functionals, denoted by , are defined by the property:

Null functionals, denoted by , are defined by the property:

In other words, moderate functionals satisfy a locally uniform polynomial estimate as when acting on , together with all derivatives, while null functionals vanish faster than any power of in the same situation. The null functionals form a differential ideal in the collection of moderate functionals.

We define space of functions and denote by . It is a subalgebra in in sense of natural identification.

Moderate functionals, denoted by , are defined by the property:

Null functionals, denoted by , are defined by the property:

Definition 1. Space of generalized functions , generalized complex numbers, and generalized real numbers is the factor algebra defined as

The space of distributions is imbedded by convolution:

Equivalence classes of sequences in will be denoted by . If is a generalized function with compact support and is a representative of , then its integral is defined by

Let , . Then

(i)they are equal in the distribution sense, if

(ii)they are associated if there exist a representative and of and , respectively, such that

3. Results on Some Products of Distributions

It was proved in [12] that for any the product of the generalized functions and in admits associated distributions and it holds

with the particular cases

for and

Here we will make a generalization of (19). In order to prove the main theorem we need the following lemmas, easily proved by induction.

Lemma 2. For one has

Lemma 3. For one has

and stands for .

Theorem 4. The product of the generalized functions and for in admits associated distributions and it holds

Proof. For given we suppose that , without loss of generality. Then using the embedding rule and the substitution we have the representatives of the distribution in Colombeau algebra:

Similar, using the embedding rule and the substitution we have the representatives of the distribution in Colombeau algebra:

Then, for any we have

using the substitution .

By the Taylor theorem we have that

for . Using this for (25) we have

where

for and we have changed the order of integration.

Putting we have

Thus

Next, suppose that is even, less than and that is less than . It follows from Lemma 2 that is an even or odd function accordingly as is odd or even. We thus have that is an odd function and . If then and by changing the order of integration we can prove that again . If we suppose that is odd and less than we can prove in a similar manner that .

For the case if again we have . For and using Lemma 3 and changing the order of integration we have

Further,

and . So, for and

Finally we have

Therefore passing to the limit, as , we obtain (22) proving the theorem.

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