An algebra of distributions on an open interval
Author:
Harris S. Shultz
Journal:
Trans. Amer. Math. Soc. 169 (1972), 163-181
MSC:
Primary 46F05
DOI:
https://doi.org/10.1090/S0002-9947-1972-0308775-4
MathSciNet review:
0308775
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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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Additional Information
Keywords:
Generalized functions,
operational calculus,
Schwartz distributions,
two-sided Laplace transformation,
Fourier transformation
Article copyright:
© Copyright 1972
American Mathematical Society