On zeros of some entire functions

Hryniv, Rostyslav; Mykytyuk, Yaroslav

doi:10.1090/S0002-9947-08-04714-4

On zeros of some entire functions
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by Rostyslav O. Hryniv and Yaroslav V. Mykytyuk PDF

Trans. Amer. Math. Soc. 361 (2009), 2207-2223 Request permission

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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