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An Inversion Formula for the Weierstrass Transform

Published online by Cambridge University Press:  20 November 2018

G. G. Bilodeau
G. G. Bilodeau*
Affiliation:
Boston College
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The Weierstrass transform f(x) of a function ϕ(y) is defined by

1.1

where

whenever this integral exists (7, p. 174). It is also known as the Gauss transform (11; 12). Its basic properties have been developed and studied in (7) and in particular it has been shown that the symbolic operator

will invert this transform under suitable assumptions and with certain definitions of this operator. We propose to study the definition

for f(x) in C∞. This formula seems to have been first examined by Pollard (9) and later by Rooney (12). In so far as convergence of (1.2) is concerned, we will considerably improve the results (12).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 13 , 1961 , pp. 593 - 601
Copyright
Copyright © Canadian Mathematical Society 1961

References

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