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Transactions of the American Mathematical Society

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On zeros of some entire functions

Authors: Rostyslav O. Hryniv and Yaroslav V. Mykytyuk
Journal: Trans. Amer. Math. Soc. 361 (2009), 2207-2223
MSC (2000): Primary 30D15; Secondary 42A38
Published electronically: November 17, 2008
MathSciNet review: 2465834
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Abstract: We study the distribution of zeros $ z_k$ for entire functions of the form $ \sin z + \int_{0}^1 f(t)\mathrm{e}^{iz(1-2t)}\,dt$ with $ f$ belonging to a space $ X \hookrightarrow L_1(0,1)$. For a large class $ \mathscr{X}$ of spaces $ X$ (including, e.g., the spaces $ L_p(0,1)$ for all $ p\in[1,\infty]$) we show that $ z_k=\pi k + \zeta_k$, where $ (\zeta_k)_{k\in\mathbb{Z}}$ is the sequence of Fourier coefficients for some function $ g$ in $ X$, and study properties of the induced mapping $ g\mapsto f$.

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Additional Information

Rostyslav O. Hryniv
Affiliation: Institute for Applied Problems of Mechanics and Mathematics, 3b Naukova st., 79601 Lviv, Ukraine – and – Lviv National University, 1 Universytetska st., 79602 Lviv, Ukraine

Yaroslav V. Mykytyuk
Affiliation: Institute for Applied Problems of Mechanics and Mathematics, 3b Naukova st., 79601 Lviv, Ukraine – and – Lviv National University, 1 Universytetska st., 79602 Lviv, Ukraine

Keywords: Entire functions, asymptotics of zeros, Fourier transform
Received by editor(s): September 26, 2006
Received by editor(s) in revised form: June 15, 2007
Published electronically: November 17, 2008
Article copyright: © Copyright 2008 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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