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On Fourier transforms of functions of the R. Nevanlinna class in the half-plane


Author: F. A. Shamoyan
Translated by: A. Plotkin
Original publication: Algebra i Analiz, tom 20 (2008), nomer 4.
Journal: St. Petersburg Math. J. 20 (2009), 665-680
MSC (2000): Primary 30D50
Published electronically: June 2, 2009
MathSciNet review: 2473749
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Abstract | References | Similar Articles | Additional Information

Abstract: Let $ f$ be a function holomorphic in the upper half-plane and belonging to the Nevanlinna class $ N(\mathbb{C}_+)$. Assume that

$\displaystyle \limsup\limits_{y \to +\infty} \frac{\ln\vert f(iy)\vert}{y} \le 0 $

and that the boundary values of $ f$ on the real axis lie in $ L^1(\mathbb{R})$. It is shown that if $ \vert\widehat{f}(x)\vert\le\frac{1}{\lambda(\vert x\vert)}$, $ x\in{\mathbb{R}_-}$, where $ \widehat{f}$ is the Fourier transform of $ f$ and $ \lambda$ is a logarithmically convex positive function on $ {\mathbb{R}_+}$, then the condition $ \int_{1}^{+\infty}\frac{\ln \lambda(x)}{x^{3/2}} dx=+\infty$ implies that $ \widehat{f}(x)=0$ for all $ x\in{\mathbb{R}_-}$. Conversely, if one of the conditions listed above fails, then there exists $ f\in N(\mathbb{C}_+) \cap L^1(\mathbb{R})$ with $ \widehat{f}(x)\ne 0$, $ x\in{\mathbb{R}_-}$.


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Additional Information

F. A. Shamoyan
Affiliation: Bryansk State University, 241050 Bryansk, Russia
Email: shamoyan@tu-bryansk.ru

DOI: http://dx.doi.org/10.1090/S1061-0022-09-01066-8
Keywords: Function of bounded characteristic, Fourier transform.
Received by editor(s): July 5, 2007
Published electronically: June 2, 2009
Article copyright: © Copyright 2009 American Mathematical Society