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On Some Properties of Functions Analytic in a Half-Plane

Published online by Cambridge University Press:  20 November 2018

P. G. Rooney
P. G. Rooney*
Affiliation:
University of Toronto
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The spaces , w real, 1 ≤ p < ∞, consist of those functions f(s), analytic for Re s > w, and such that μp(f;x) is bounded for x > w, where

1.1

Doetsch (1) has shown that if e-wtϕ(t) ∈ Lp (0, ∞), 1 < p ≤ 2, and f is the Laplace transform of ϕ, that is,

then f ∈ , where

1.2

and that conversely if f ∈ , 1 < p ≤ 2, then there is a function ϕ, with e-wtϕ(t) ∈ Lq (0, ∞), such that f is the Laplace transform of ϕ.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 11 , 1959 , pp. 432 - 439
Copyright
Copyright © Canadian Mathematical Society 1959

References

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