Some inequalities for entire functions
Анотація
Let Lp be the subspace of the space Lp(R) consisting of the restriction to the real axis of all entire functions of exponential type ≤π. In this paper, for any function f∈Lp(1≤p≤∞), we obtain estimates for the norm of f in terms of the sequence (f(n/2))n∈Z, namely
12‖
where \|f\|_{p,1}:=\frac12(\|Jf\|_{\ell_p(\mathbb{Z})} +\|JT_{1/2}f\|_{\ell_p(\mathbb{Z})}). Here J:\mathcal{L}_p\to\ell_p(\mathbb{Z}) is the linear operator given by the formula (Jf)(n):=(-1)^nf(n), \quad n\in\mathbb{Z}, and T_\tau is the shift by \tau\in\mathbb{R} of the function f,
(T_\tau f)(z):=f(z+\tau), \quad z\in\mathbb{C}.
Посилання
B.Ya. Levin, Lectures on entire functions, Translations of Mathematical Monographs, V.150, AMS, Providence, RI, 1996.
I.S. Gradshteyn, I.M. Ryzhik, Table of integrals, series, and products, Academic Press, Editors: Daniel Zwillinger, Victor Moll, 2014. doi.org/10.1016/C2010-0-64839-5
Авторське право (c) 2024 D. Lukivska, N. Sushchyk

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