Colombeau products of distributions

In this paper, some products of distributions are derived. The results are obtained in Colombeau algebra of generalized functions, which is the most relevant algebraic construction for dealing with Schwartz distributions. Colombeau algebra \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{G}}({\mathbf{R}})$$\end{document}G(R) contains the space \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal{D}'}({\mathbf{R}})$$\end{document}D′(R) of Schwartz distributions as a subspace, and has a notion of 'association’ that allows us to evaluate the results in terms of distributions.

By the regularization process, the nonlinear structure is lost in a way by identifying sequences with their limit. Actually, all the operations then are done on the regularized functions (sequences of smooth functions) and with the inverse process starting from the result, the function is returned from the regularization. So, we have to get nonlinear theory of generalized functions that will work with regularization.
The optimal solution for overcoming the problems that Schwartz theory of distributions is concerned with was offered by Colombeau (1984Colombeau ( , 1985. He constructed an associative differential algebra of generalized functions G(R), which contains the space D ′ (R) of distributions as subspace and the algebra of C ∞ -functions as subalgebra. This theory of generalized functions of Colombeau actually generalizes the theory of Schwartz distributions: these new Colombeau generalized functions can be differentiated in the same way as distributions, but where multiplication and other nonlinear operations are concerned, it is significant that the result of these operations always exists in this algebra as Colombeau generalized function [How Colombeau algebra G can be used for treating linear and nonlinear problems, including singularities, one can see in Jolevska et al. (2007)]. 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 the Schwartz definition of distributions.
The notion of 'association' in G is a faithful generalization of the equality of distributions, and again enables us to interpret results in terms of distributions.
Due to all these properties, Colombeau theory has found extensive application in different natural sciences and engineering, especially in fields where products of distributions with coinciding singularities are considered. Such products are, for example, products that include Dirac delta function δ which has singular point support.
Before distribution theory come to be introduced, Dirac delta function, as with many other concepts in physics and engineering, was heuristically understood with properties given to coincide with experimental results and was considered adequate for solving complicated problems. Delta function δ was well defined to represent certain types of infinity concentrated at a single point (by the physicists δ was used to represent the charge density of a point particle-the charge is concentrated in a single point, i.e., finite amount of charge is packed into zero volume, thus the charge density must be infinite at that single point, and its derivative δ ′ was used to represent a dipole of unit electric moment at the origin). After the theory of distributions was invented (in the early 1950s), the mathematical meaning of these concepts has been established and delta function with the same properties is considered as distribution. But, the problem still occurs when multiplying two distributions in the Schwartz's space (for example, δ 2 doesn't exist in this space). As we said above, Colombeau algebra was constructed in a way that many problems with multiplication of distributions could be avoided. About applications of Colombeau theory of generalized functions, one can read papers (Aragona et al. 2014;Gsponer 2009;Ohkitani and Dowker 2010;Prusa and Rajagopal 2016;Steinbauer and Vickers 2006;Capar 2013;Nigsch and Samman 2013;Steinbauer 1997;Alimohammady 2014;Sojanovic 2013;Farassat 1994). As we can see in many of these papers, products of delta function and its derivatives with other distributions with singularities, but with continuous functions too, appear while solving various problems in physics and engineering.
In this paper we obtain some products that include derivatives of delta function, in Colombeau algebra, in terms of associated distributions. Other products of distributions, evaluated in the same way, can be found in Damyanov (1997Damyanov ( , 2005Damyanov ( , 2006, Miteva and Jolevska (2012), Jolevska and Atanasova (2013) and Miteva et al. (2014). The results can be reformulated as regularized products in the classical distribution theory.

Colombeau algebra
In this section we will give notations and definitions from Colombeau theory that we have used while evaluating the main results.
Colombeau in his books has proved that the sets A k are non empty for all k ∈ N.
For ϕ ∈ A q (R) and ε > 0 it is denoted as is the subalgebra of 'moderate' functions such that for each compact subset K of R and any p ∈ N 0 there is a q ∈ N, such that for each ϕ ∈ A q (R) there are c > 0, η > 0 and it holds: consisting of all functions f (ϕ, x) such that for each compact subset K of R and any p ∈ N 0 there is a q ∈ N such that for every r ≥ q and each ϕ ∈ A r (R) there are c > 0, η > 0 and it holds: The distributions on R are embedded in the Colombeau algebra G(R) by the map: where * denotes the convolution product of two distributions and is given by: We should notice that the sequential approach (regularization method) mentioned in the previous section is used here. Thus, an element f ∈ G (a generalized function of Colombeau) is actually an equivalence class [f ] = [f ε + I] of an element f ε ∈ E M which is called representative of f. Multiplication and differentiation of generalized functions are performed on arbitrary representatives of the respective generalized functions.
The meaning of the term 'association' in G(R) is given with the next two definitions.
Definition 1 Generalized functions f , g ∈ G(R) are said to be associated, denoted f ≈ g, if for each representative f (ϕ ε , x) and g(ϕ ε , x) and arbitrary ψ(x) ∈ D(R) there is a q ∈ N 0 such that for any ϕ(x) ∈ A q (R) x) of f and any ψ(x) ∈ D(R) there is a q ∈ N 0 such that for any ϕ(x) ∈ A q (R) The representatives chosen in the above two definitions don't affect the result. The distribution associated, if it exists, is unique and the association is a faithful generalization of the equality of distributions.
If we multiply two distributions embedded in G, as a result we always obtain a generalized function of Colombeau. But, it may not always be associated to a third distribution, so if the product of two distributions embedded in Colombeau algebra G admits an associated distribution, we say that Colombeau product of those two distributions exists. If the regularized model product of two distributions exists, then their Colombeau product also exists and it is the same as the first one.

Main results
Theorem 1 The product of the generalized functions (cos x − sin x) and δ (r) (x) for r = 0, 1, 2, . . . in G(R) admits associated distribution and it holds: is the floor function.
In a similar way we obtain the embedding of the distribution δ (r) (x) in Colombeau algebra: Then, for any ψ(x) ∈ D(R) we have: where we have used substitution u = − x ε . Applying Taylor's Theorem for the function ψ we have: for 0 < η < 1. Using (16) in (15) and changing the order of integration we obtain: for i, j = 0, 1, 2, . . . r. Now using binomial expansion, for the last integral we obtain: The integral J a,b = l −l v a ϕ (b) (v)dv is nonzero only for a = b and its value is J a,a = (−1) a a! . Thus the only nonzero term in the above sum is obtained for k = 0 and i + j = r, and the value J i,j then will be and Eq. (19) in (17) we obtain: 1+ i 2 and passing to the limit, as ε → 0, we obtain (10), which proves the Theorem 1.

Theorem 2
The product of the generalized functions (sin x + cos x) and δ (r) (x) for r = 0, 1, 2, . . . in G(R) admits associated distribution and it holds: Proof Applying Taylor's Theorem for the functions sin x and cos x we have: where a i = i 2 . Now following the proof of the previous theorem, using the same steps, we obtain (21).

Theorem 3
The product of the generalized functions e x and δ (r) (x) for r = 0, 1, 2, . . . in G(R) admits associated distribution and it holds: Proof Expanding function e x in a Taylor series we have: Thus if we take a i = 0 for i = 0, 1, 2 . . . in the proof of the Theorem 1 we obtain (23).
Remark We should notice here that the products (cos x ± sin x) · δ (r) (x) and e x · δ (r) (x) make sense even in the classical setting (Schwartz space of distributions), since multiplication of a classical distribution by a smooth function is valid operation in the Schwartz theory, but we have obtained here results in terms of associated distributions, which is a faithful generalization of the 'weak' equality in G(R). The results obtained in our theorems are associated with terms consisting only of the delta function and its derivatives. We will also notice that the product sin x · δ(x) is not equal to zero in G(R) if we consider the 'strong' equality, but it is zero in the sense of association, i.e. in the sense of 'weak' equality in G(R).

Conclusion
We have evaluated some products of generalized functions, involving derivatives of the Dirac delta function, in Colombeau algebra in terms of associated distributions. This is significant because products of this type are very often used not only in physics, especially in quantum physics, but in other natural sciences and engineering, too, as we can see in the cited literature. Colombeau differential algebra of generalized functions contains the space of Schwartz distributions as a subspace, and the product of elements in it is generalization of the product of distributions, and thus all the results obtained in this way can be reformulated as regularized products in the classical distribution theory.