An algebra of distributions on an open interval

Shultz, Harris S.

doi:10.1090/S0002-9947-1972-0308775-4

An algebra of distributions on an open interval
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by Harris S. Shultz PDF

Trans. Amer. Math. Soc. 169 (1972), 163-181 Request permission

Abstract:

Let $(a,b)$ be any open subinterval of the reals which contains the origin and let $\mathfrak {B}$ denote the family of all distributions on $(a,b)$ which are regular in some interval $( \in ,0)$, where $\in < 0$. Then $\mathfrak {B}$ is a commutative algebra: Multiplication is defined so that, when restricted to those distributions on $(a,b)$ whose supports are contained in $[0,b)$, it is ordinary convolution. Also, $\mathfrak {B}$ can be injected into an algebra of operators; this family of operators is a sequentially complete locally convex space. Since it preserves multiplication, this injection serves as a generalization (there are no growth restrictions) of the two-sided Laplace transformation.

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Initial-value problems involving generalized functions; two-sided operational calculus

A new algebra of distributions; initial-value problems involving Schwartz distributions

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