# Direct Products and Convolutions of Distributions

• Ram P. Kanwal

## Abstract

Let R m and R n be Euclidean spaces of dimensions m and n respectively, and let x = (x 1,…,x m ) and y = (y 1,…,y n ) denote the generic points in R m and R n ), respectively. Then a point in the Cartesian product R m +n = R m x R n is (x,y) = (x 1,…, x m , y 1,…, y n ). Furthermore, let us denote by D m , D n , and D m +n the spaces of test functions with compact support in R m ,R n , andR m +n, respectively. When f (x ) and g(y) are locally integrable functions in the spaces R m and R n , then the function f(x)g(y) is also locally integrable function in R m +n. It defines the regular distribution:
$$\begin{array}{*{20}c} {\left\langle {f\left( x \right)g\left( y \right),\varphi \left( {x,y} \right)} \right\rangle = \int {f\left( x \right)} \int {g\left( y \right)\varphi \left( {x,y} \right)dy dx} } \\ { = \left\langle {f\left( x \right),\left\langle {g\left( y \right),\varphi \left( {x, y} \right)} \right\rangle } \right\rangle } \\ \end{array}$$
(1)
or
$$\begin{array}{*{20}c} {\left\langle {g(y)f(x),\varphi (x,y)} \right\rangle = \int {g(y)} \int {f(x)\varphi (x,y)dx dy} } \\ { = \left\langle {g(y),\left\langle {f(x),\varphi (x,y)} \right\rangle } \right\rangle } \\ \end{array}$$
(2)
for φ (x, y) ∈ Dm+n. Let us denote by s(x) ⊗ t(y) the direct product of the distributions s(x) ∈ D m and t(y) ∈ D n according to (1),
$$\left\langle {s(x) \otimes t(y),\varphi (x,y)} \right\rangle = \left\langle {s(x),\left\langle {t(y),\varphi (x,y)} \right\rangle } \right\rangle , \varphi (x,y) \in D_{m + n,}$$
(3)
and check whether the right side of this equation defines a linear continuous functional over Dm+n . For this purpose , we prove the following lemma: Lemma 1. The function ψ (x) = 〈t(y), φ(x, y)〉, where t ∈ D n and φ(x, y) ∈ Dm+n, is a testfunction in Dm, and
$$D^k \psi (x) = \left\langle {t(y),D_x^k \varphi (x,y)} \right\rangle$$
(4)
for all multiindices k, where D x k implies differentiation with respect to (X1, x2,..., xm) only. Also, if the sequence {φ (x, y)} → φ (x, y) in Dm+n as l → ∞, then the sequence ψl (x) = {〈t( y), φ(x, y)〉} → ψ(x) in Dm as l → ∞.

## Keywords

Integral Equation Direct Product Compact Support Integrable Function Abel Integral Equation
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

## Preview

Unable to display preview. Download preview PDF.