Geodesic
On Stable Reconstruction of Analytic Functions from Fourier Samples
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 26 (2020) no. 4, pp. 182-195 Cet article a éte moissonné depuis la source Math-Net.Ru

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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 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: $\kappa_{\delta_m}$. 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 $\kappa_{\delta_m} \ge c\,m^{1/2}$.
Mots-clés : Fourier coefficients, stable reconstruction
Keywords: polynomial inequalities. 
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     title = {On {Stable} {Reconstruction} of {Analytic} {Functions} from {Fourier} {Samples}},
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S. V. Konyagin; A. Yu. Shadrin. On Stable Reconstruction of Analytic Functions from Fourier Samples. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 26 (2020) no. 4, pp. 182-195. http://geodesic.mathdoc.fr/item/TIMM_2020_26_4_a11/

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