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JOURNALS // Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya // Archive

Izv. RAN. Ser. Mat., 2017 Volume 81, Issue 6, Pages 23–37 (Mi im8529)

This article is cited in 10 papers

Approximation by sums of shifts of a single function on the circle

P. A. Borodin

Lomonosov Moscow State University

Abstract: We study approximation properties of the sums $\sum_{k=1}^nf(t-a_k)$ of shifts of a single function $f$ in real spaces $L_p(\mathbb{T})$ and $C(\mathbb{T})$ on the circle $\mathbb{T}=[0,2\pi)$, and also in complex spaces of functions analytic in the unit disc. We obtain sufficient conditions in terms of the trigonometric Fourier coefficients of $f$ for these sums to be dense in the corresponding subspaces of functions with zero mean. We investigate the accuracy of these conditions. We also suggest a simple algorithm for the approximation by sums of plus or minus shifts of one particular function in $L_2(\mathbb{T})$ and obtain bounds for the rate of approximation.

Keywords: approximation, sums of shifts, Fourier coefficients, semigroup.

UDC: 517.518.843+517.982.256

MSC: 41A30, 41A25

Received: 18.02.2016
Revised: 21.08.2016

DOI: 10.4213/im8529


 English version:
Izvestiya: Mathematics, 2017, 81:6, 1080–1094

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© Steklov Math. Inst. of RAS, 2025