This article is cited in 2 papers

Completeness and Basis Properties of Systems of Exponentials in Weighted Spaces $L^p(-\pi,\pi)$

A. A. Yukhimenko

M. V. Lomonosov Moscow State University

Abstract: We consider the system of exponentials $e(\Lambda)=\{e^{i\lambda_nt}\}_{n\in\mathbb Z}$, where
$$ \lambda_n=n+\biggl(\frac{1+\alpha}p+l(|n|)\biggr)\operatorname{sign}n, $$
$l(t)$ is a slowly varying function, and $l(t)\to 0$, $t\to\infty$. We obtain an estimate for the generating function of the sequence $\{\lambda_n\}$ and, with its help, find a completeness criterion and a basis condition for the system $e(\Lambda)$ in the weight spaces $L^p(-\pi,\pi)$. We also study some special cases of the function $l(t)$.

Keywords: system of exponentials, completeness of a system of functions, the weight spaces $L^p(-\pi,\pi)$, Laplace transform, Cauchy's theorem, Riesz basis, generating function.

UDC: 517.5

Received: 27.02.2006
Revised: 10.07.2006

DOI: 10.4213/mzm3723

 English version: Mathematical Notes, 2007, 81:5, 695–707

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