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On Some Properties of Functions Analytic in a Half-Plane
Published online by Cambridge University Press: 20 November 2018
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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.
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- Copyright © Canadian Mathematical Society 1959
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