Direct Products and Convolutions of Distributions

Ram P. Kanwal
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Ram P. Kanwal: The Pennsylvania State University, Department of Mathematics

Chapter Chapter 7 in Generalized Functions Theory and Technique, 1998, pp 173-207 from Springer

Abstract: Abstract Let R m and R n be Euclidean spaces of dimensions m and n respectively, and let x = (x1,..., x m ) and y = (y1,..., 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 × R n is (x, y) = (x1,..., x m , y1,..., 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 , and Rm+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: 1 $$\left\langle f\left( x \right)g\left( y \right),\phi \left( x,y \right) \right\rangle _{=\left\langle f\left( x \right),\left\langle g\left( y \right),\phi \left( x,y \right) \right\rangle \right\rangle }^{=\int{f\left( x \right)\int{g\left( y \right)\phi \left( x,y \right)dy\text{ }dx}}}$$ or 2 $$\left\langle g\left( y \right)f\left( x \right),\phi \left( x,y \right) \right\rangle _{=\left\langle g\left( y \right),\left\langle f\left( x \right),\phi \left( x,y \right) \right\rangle \right\rangle }^{=\int{g\left( y \right)\int{f\left( x \right)\phi \left( x,y \right)dx\text{ }dy}}}$$ for φ(x,y) ∈ D m+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), 3 $$\left\langle s\left( x \right)\otimes t\left( y \right),\phi \left( x,y \right) \right\rangle =\left\langle s\left( x \right),\left\langle t\left( y \right),\phi \left( x,y \right) \right\rangle \right\rangle ,\text{ }\phi \left( x,y \right)\in {{D}_{m+n}},$$ and check whether the right side of this equation defines a linear continuous functional over D m+n . For this purpose, we prove the following lemma: Lemma 1.The function ψ(x) = , where t ∈ D′ n and ψ(x,y) ∈ D m+n , is a test function in D m , and 4 $${{D}^{k}}\psi \left( x \right)=\left\langle t\left( y \right),D_{X}^{K}\phi \left( x,y \right) \right\rangle$$ For all multiindices k, where D x k implies differentiation with respect to (x1, x2,..., x m ) only. Also, if the sequence {φ l (x,y)} → φ(x,y) in D m+n as l → ∞, then the sequence ψl(x) = { } → ψ(x) in D m as l → ∞.

Keywords: Integral Equation; Direct Product; Compact Support; Integrable Function; Distributional Solution (search for similar items in EconPapers)
Date: 1998
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DOI: 10.1007/978-1-4684-0035-9_7

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