Trudy Instituta Matematiki i Mekhaniki UrO RAN
RUS  ENG    JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB  
General information
Latest issue
Archive
Impact factor

Search papers
Search references

RSS
Latest issue
Current issues
Archive issues
What is RSS



Trudy Inst. Mat. i Mekh. UrO RAN:
Year:
Volume:
Issue:
Page:
Find






Personal entry:
Login:
Password:
Save password
Enter
Forgotten password?
Register


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

Full text: PDF file (235 kB)
References: PDF file   HTML file

Bibliographic databases:

UDC: 519.651 + 517.518.454 + 517.518.86
MSC: 65D15, 41A10, 41A17, 42A16
Received: 29.06.2020
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}


Linking options:
  • http://mi.mathnet.ru/eng/timm1774
  • http://mi.mathnet.ru/eng/timm/v26/i4/p182

    SHARE: VKontakte.ru FaceBook Twitter Mail.ru Livejournal Memori.ru


    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles
  • Trudy Instituta Matematiki i Mekhaniki UrO RAN
    Number of views:
    This page:36
    Full text:9
    References:1
    First page:5

     
    Contact us:
     Terms of Use  Registration to the website  Logotypes © Steklov Mathematical Institute RAS, 2021