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Representation Formulas for Integrable and Entire Functions of Exponential Type I

Published online by Cambridge University Press:  20 November 2018

Clément Frappier
Clément Frappier*
Affiliation:
Université de Montréal, Montréal, Québec
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Let Bτ denote the class of entire functions of exponential type τ (>0) bounded on the real axis. For the function f ∊ Bτ we have the interpolation formula [1, p. 143]

1.1

where t, γ are real numbers and is the so called conjugate function of f. Let us put

1.2

The function Gγ,f is a periodic function of α, with period 2. For t = 0 (the general case is obtained by translation) the righthand member of (1) is 2τGγ,f (1). In the following paper we suppose that f satisfies an additional hypothesis of the form f(x) = O(|x|-ε), for some ε > 0, as x → ±∞ and we give an integral representation of Gγ,f(α) which is valid for 0 ≦ α ≦ 2.

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 40 , Issue 4 , 01 August 1988 , pp. 1010 - 1024
Copyright
Copyright © Canadian Mathematical Society 1988

References

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