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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



An algebra of distributions on an open interval

Author: Harris S. Shultz
Journal: Trans. Amer. Math. Soc. 169 (1972), 163-181
MSC: Primary 46F05
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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Keywords: Generalized functions, operational calculus, Schwartz distributions, two-sided Laplace transformation, Fourier transformation
Article copyright: © Copyright 1972 American Mathematical Society