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 \(\delta_{m}=\{j:|j|\leq m\}\), 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 the sets \((\delta_{m})\) are chosen freely, there exist reconstruction operators \((\phi_{\delta_{m}})\) that have exponential convergence rate and are almost stable; specifically, their condition numbers grow at most linearly: \(\kappa_{\delta_{m}}<c\,m\). 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 \(\kappa_{\delta_{m}}\geq c\,m^{1/2}\).
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The work of the first author was supported by a grant of the Government of the Russian Federation (project no. 14.W03.31.0031).
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Translated from Trudy Instituta Matematiki i Mekhaniki UrO RAN, Vol. 26, No. 4, pp. 182 - 195, 2020 https://doi.org/10.21538/0134-4889-2020-26-4-182-195.
Translated by M. Deikalova
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Konyagin, S.V., Shadrin, A.Y. On Stable Reconstruction of Analytic Functions from Fourier Samples. Proc. Steklov Inst. Math. 315 (Suppl 1), S178–S191 (2021). https://doi.org/10.1134/S0081543821060146
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DOI: https://doi.org/10.1134/S0081543821060146