Entire functions of exponential type, almost periodic in Besicovitch's sense on the real hyperplane

S.Yu.Favorov; O.I.Udodova

Suppose that an almost periodic in Besicovitch's sense function $f(x)$ on $\mathbb{R}^p$ is the restriction to $\mathbb{R}^p$ of some entire function of exponential type $\sigma$ in $\mathbb{C}^p$. Then the spectrum of $f$ is contained in the ball $\{x~\in~\mathbb{R}^p:~|x|~\le~\sigma~ \}$.

1. Besicovitch A.S. Almost periodic functions. -- Cambridge university press, 1932. -- 253 p.

2. Boas R. P. Jr. Functions of exponential type. I. // Duke Math. J. -- 1944. -- P.~9--85.

3. Bohr H. Zur Theorie der fastperiodischen Funktionen. III Teil. Dirichlet-entwicklung analytischer Funktionen // Acta Math. -- 1926. -- V.~44. -- P.~237--281.

4. Girya N.P. Almost periodic in Besicovitch's metric functions with the spectrum in a cone // Matematychni Studii. -- 2007. -- V.27, №2. -- P.~162--173.

5. Levin B. Ya. Distribution of zeros of entire functions. -- М.: GITTL. -- 1956. -- 639~p. (in Russian)

6. Udodova O.I. Holomorphic almost periodic functions in various metrics // Vestnik of Kharkov National University. Ser. "Mathematics, Applied Mathematics, and Mechanics". -- 2003. -- V.~52, №589. -- P.~90--107. (in Russian)

7. Udodova O.I. Fourier series of holomorphic almost periodic in Besicovitch's sense functions. // Vestnik of Kharkov National University. Ser. "Mathematics, Applied Mathematics, and Mechanics". -- 2004. -- V.~53, №645. -- P.~53--64. (in Russian)

8. Favorov S.Yu., Udodova O.I. Almost periodic functions in finite--dimensional space with the spectrum in a cone // Math. Physics, Analys, Geometry. -- 2002. -- V.~9, №3. -- P.~465--477.

9. Logvinenko V. N. On one multidimensional generalization of Cartwright's Theorem // Dokl. AN SSSR. -- 1974. -- V.~219, №3. -- P.~546--549. (in Russian)

	Entire functions of exponential type, almost periodic in Besicovitch's sense on the real hyperplane
Author	S.Yu.Favorov, O.I.Udodova Sergey.Ju.Favorov@univer.kharkov.ua, udodova@kart.edu.ua Department of Mechanics and Mathematics, Kharkov National,University,61077 Kharkov, Svobody sq. 4, Ukraine, Department of Mathematics, Ukrainian State Academy of Railway,Transport,61050 Kharkov, Feyerbakh sq. 7, Ukraine
Abstract	Suppose that an almost periodic in Besicovitch's sense function $f(x)$ on $\mathbb{R}^p$ is the restriction to $\mathbb{R}^p$ of some entire function of exponential type $\sigma$ in $\mathbb{C}^p$. Then the spectrum of $f$ is contained in the ball $\{x~\in~\mathbb{R}^p:~\|x\|~\le~\sigma~ \}$.
Keywords	almost periodic function, Besicovitch's sense, entire function, exponential type, spectrum
DOI	doi:10.30970/ms.30.2.147-152
Reference	1. Besicovitch A.S. Almost periodic functions. -- Cambridge university press, 1932. -- 253 p. 2. Boas R. P. Jr. Functions of exponential type. I. // Duke Math. J. -- 1944. -- P.~9--85. 3. Bohr H. Zur Theorie der fastperiodischen Funktionen. III Teil. Dirichlet-entwicklung analytischer Funktionen // Acta Math. -- 1926. -- V.~44. -- P.~237--281. 4. Girya N.P. Almost periodic in Besicovitch's metric functions with the spectrum in a cone // Matematychni Studii. -- 2007. -- V.27, №2. -- P.~162--173. 5. Levin B. Ya. Distribution of zeros of entire functions. -- М.: GITTL. -- 1956. -- 639~p. (in Russian) 6. Udodova O.I. Holomorphic almost periodic functions in various metrics // Vestnik of Kharkov National University. Ser. "Mathematics, Applied Mathematics, and Mechanics". -- 2003. -- V.~52, №589. -- P.~90--107. (in Russian) 7. Udodova O.I. Fourier series of holomorphic almost periodic in Besicovitch's sense functions. // Vestnik of Kharkov National University. Ser. "Mathematics, Applied Mathematics, and Mechanics". -- 2004. -- V.~53, №645. -- P.~53--64. (in Russian) 8. Favorov S.Yu., Udodova O.I. Almost periodic functions in finite--dimensional space with the spectrum in a cone // Math. Physics, Analys, Geometry. -- 2002. -- V.~9, №3. -- P.~465--477. 9. Logvinenko V. N. On one multidimensional generalization of Cartwright's Theorem // Dokl. AN SSSR. -- 1974. -- V.~219, №3. -- P.~546--549. (in Russian)
Pages	147-152
Volume	30
Issue	2
Year	2008
Journal	Matematychni Studii
Full text of paper	pdf
Table of content of issue	html