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Mat. Sb., 2019, Volume 210, Number 4, Pages 145–164 (Mi msb8890)  

Equivalence of the trigonometric system and its perturbations in the spaces $L^p$ and $C$

A. M. Sedletskii

Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow, Russia

Abstract: Let $B=B[-\pi,\pi]$ be any of the spaces $L^p(-\pi,\pi)$, $1\leq p<\infty$, $p\neq2$, and $C[-\pi,\pi]$, and let $B_a=B[-\pi+a,\pi+a]$, $a\in\mathbb R$. A number of necessary conditions and sufficient conditions for the ‘perturbed trigonometric system’ $e^{i(n+\alpha_n)t}$, $n\in\mathbb Z$, to be equivalent to the trigonometric system $e^{int}$, $n\in\mathbb Z$, in the space $B_a$ for any $a\in\mathbb R$ are obtained. In particular, it is shown that if $(\alpha_n)\in l^s$, where $1/s=|1/p-1/2|$, then this equivalence takes place, the exponent $s$ being sharp. This result is used to show that in $L^p(-\pi,\pi)$, $1<p<2$, there exist bases of exponentials which are not equivalent to the trigonometric basis.
The machinery of Fourier multipliers is used in the proofs.
Bibliography: 18 titles.

Keywords: equivalent systems of functions, basis, Fourier multiplier.

Funding Agency Grant Number
Lomonosov Moscow State University
This research was carried out with the financial support of Lomonosov Moscow State University (grant “Current problems in fundamental mathematics and mechanics”).


DOI: https://doi.org/10.4213/sm8890

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English version:
Sbornik: Mathematics, 2019, 210:4, 606–624

Bibliographic databases:

UDC: 517.982.254
MSC: 46B15
Received: 21.12.2016 and 02.09.2018

Citation: A. M. Sedletskii, “Equivalence of the trigonometric system and its perturbations in the spaces $L^p$ and $C$”, Mat. Sb., 210:4 (2019), 145–164; Sb. Math., 210:4 (2019), 606–624

Citation in format AMSBIB
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