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 Trudy Inst. Mat. i Mekh. UrO RAN, 2020, Volume 26, Number 4, Pages 182–195 (Mi timm1774)

On stable reconstruction of analytic functions from Fourier samples

S. V. Konyaginab, A. Yu. Shadrinc

a Lomonosov Moscow State University, Faculty of Mechanics and Mathematics
b Steklov Mathematical Institute of Russian Academy of Sciences, Moscow
c University of Cambridge, Department of Applied Mathematics and Theoretical Physics

Abstract: Stability of reconstruction of analytic functions from the values of $2m+1$ coefficients of its Fourier series is studied. The coefficients can be taken from an arbitrary symmetric set $\delta_m \subset \mathbb{Z}$ of cardinality $2m+1$. It is known that, for $\delta_m=\{ j: |j| \le m\}$, i.e., if the coefficients are consecutive, the fastest possible convergence rate in the case of stable reconstruction is an exponential function of the square root of $m$. Any method with faster convergence is highly unstable. In particular, exponential convergence implies exponential ill-conditioning. In this paper, we show that, if we are free to choose any sets $(\delta_m)$, there exist reconstruction operators $(\phi_{\delta_m})$ that have exponential convergence rate and are almost stable; specifically, their condition numbers grow at most linearly: $\kappa_{\delta_m}<c \cdot m$. We also show that this result cannot be noticeably strengthened. More precisely, for any sets $(\delta_m)$ and any reconstruction operators $(\phi_{\delta_m})$, exponential convergence is possible only if $\kappa_{\delta_m} \ge c \cdot m^{1/2}$.

Keywords: Fourier coefficients, stable reconstruction, polynomial inequalities.

 Funding Agency Grant Number Ministry of Science and Higher Education of the Russian Federation 14.W03.31.0031 The work of the first author was supported by a grant of the Government of the Russian Federation (project no. 14.W03.31.0031).

DOI: https://doi.org/10.21538/0134-4889-2020-26-4-182-195

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Bibliographic databases:

UDC: 519.651 + 517.518.454 + 517.518.86
MSC: 65D15, 41A10, 41A17, 42A16
Revised: 10.10.2020
Accepted:19.10.2020

Citation: S. V. Konyagin, A. Yu. Shadrin, “On stable reconstruction of analytic functions from Fourier samples”, Trudy Inst. Mat. i Mekh. UrO RAN, 26, no. 4, 2020, 182–195

Citation in format AMSBIB
\Bibitem{KonSha20} \by S.~V.~Konyagin, A.~Yu.~Shadrin \paper On stable reconstruction of analytic functions from Fourier samples \serial Trudy Inst. Mat. i Mekh. UrO RAN \yr 2020 \vol 26 \issue 4 \pages 182--195 \mathnet{http://mi.mathnet.ru/timm1774} \crossref{https://doi.org/10.21538/0134-4889-2020-26-4-182-195} \elib{https://elibrary.ru/item.asp?id=44314667}