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 Izv. RAN. Ser. Mat., 2017, Volume 81, Issue 6, Pages 23–37 (Mi izv8529)

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.

 Funding Agency Grant Number Russian Foundation for Basic Research 14-01-0051015-01-08335 Dynasty Foundation This paper was written with the financial support of RFBR (grants nos. 14-01-00510, 15-01-08335) and the Dmitry Zimin Dynasty Foundation.

DOI: https://doi.org/10.4213/im8529

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

Bibliographic databases:

UDC: 517.518.843+517.982.256
MSC: 41A30, 41A25
Revised: 21.08.2016

Citation: P. A. Borodin, “Approximation by sums of shifts of a single function on the circle”, Izv. RAN. Ser. Mat., 81:6 (2017), 23–37; Izv. Math., 81:6 (2017), 1080–1094

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/izv8529
• https://doi.org/10.4213/im8529
• http://mi.mathnet.ru/eng/izv/v81/i6/p23

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. P. A. Borodin, S. V. Konyagin, “Convergence to zero of exponential sums with positive integer coefficients and approximation by sums of shifts of a single function on the line”, Anal. Math., 44:2 (2018), 163–183
2. P. A. Borodin, “Approximation by Sums of the Form $\sum_k\lambda_kh(\lambda_kz)$ in the Disk”, Math. Notes, 104:1 (2018), 3–9
3. P. A. Borodin, “Density of sums of shifts of a single vector in sequence spaces”, Proc. Steklov Inst. Math., 303 (2018), 31–35
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