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Mat. Sb., 2012, Volume 203, Number 3, Pages 49–78 (Mi msb7797)  

This article is cited in 14 scientific papers (total in 14 papers)

Modifications of functions, Fourier coefficients and nonlinear approximation

M. G. Grigoryan

Yerevan State University

Abstract: This work continues the author's investigations of the convergence of greedy algorithms from the standpoint of classical results on correction of functions. In particular, the following result is obtained: for each $\varepsilon$, $0<\varepsilon<1$, there exists a measurable set $E\subset [0,1)$ of measure $|E|>1-\varepsilon$ such that for each function $f\in L^{1}[0,1)$ a function $\widetilde{f}\in L^{1}(0,1)$ equal to $f$ on $E$ can be found such that the greedy algorithm for $\widetilde{f}$ with respect to the Walsh system converges to it almost everywhere on $[0,1]$, and all the nonzero elements of the sequence of Walsh-Fourier coefficients of the function thus obtained are arranged in decreasing order of their absolute values.
Bibliography: 35 titles.

Keywords: Fourier coefficients, correction of functions, nonlinear approximation, greedy algorithm.

DOI: https://doi.org/10.4213/sm7797

Full text: PDF file (657 kB)
References: PDF file   HTML file

English version:
Sbornik: Mathematics, 2012, 203:3, 351–379

Bibliographic databases:

UDC: 517.518.454+517.518.36+517.518.8
MSC: 42C10
Received: 08.10.2010 and 20.04.2011

Citation: M. G. Grigoryan, “Modifications of functions, Fourier coefficients and nonlinear approximation”, Mat. Sb., 203:3 (2012), 49–78; Sb. Math., 203:3 (2012), 351–379

Citation in format AMSBIB
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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. M. G. Grigoryan, S. A. Sargsyan, “Nonlinear approximation of functions from the class $L^r$ with respect to the Vilenkin system”, Russian Math. (Iz. VUZ), 57:2 (2013), 25–33  mathnet  crossref
    2. M. Grigoryan, A. Minasyan, “Representation of functions in $L_\mu^1$ weighted spaces by series with monotone coefficients in the Walsh genrealized system”, Appl. Math., 4:11 (2013), 6–11  crossref
    3. A. B. Minasyan, “On Fourier coefficients with respect to the Walsh double system”, Uch. zapiski EGU, ser. Fizika i Matematika, 2014, no. 1, 22–25  mathnet
    4. L. N. Galoyan, M. G. Grigoryan, A. Kh. Kobelyan, “Convergence of Fourier series in classical systems”, Sb. Math., 206:7 (2015), 941–979  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    5. Grigoryan M.G., “Nonlinear approximation by the trigonometric system in weighted $L_\mu^p$ spaces”, J. Contemp. Math. Anal., 50:3 (2015), 128–140  crossref  mathscinet  zmath  isi  scopus
    6. L. N. Galoyan, R. G. Melikbekyan, “Behavior of the Fourier–Walsh coefficients of a corrected function”, Siberian Math. J., 57:3 (2016), 505–512  mathnet  crossref  crossref  mathscinet  isi  elib  elib
    7. M. G. Grigoryan, K. A. Navasardyan, “Universal functions in ‘correction’ problems guaranteeing the convergence of Fourier–Walsh series”, Izv. Math., 80:6 (2016), 1057–1083  mathnet  crossref  crossref  mathscinet  adsnasa  isi  elib
    8. M. G. Grigoryan, K. A. Navasardyan, “On behavior of Fourier coefficients by Walsh system”, J. Contemp. Math. Anal. Armen. Aca., 51:1 (2016), 21–33  crossref  zmath  isi  scopus
    9. M. G. Grigoryan, S. Sargsyan, “On the Fourier-Vilenkin coefficients”, Acta Math. Sci. Ser. B Engl. Ed., 37:2 (2017), 293–300  crossref  mathscinet  zmath  isi
    10. M. G. Grigoryan, “On the universal and strong $(L^1,L^\infty)$-property related to Fourier-Walsh series”, Banach J. Math. Anal., 11:3 (2017), 698–712  crossref  mathscinet  zmath  isi  scopus
    11. Grigoryan M.G. Sargsyan S.A., “On the l1-Convergence and Behavior of Coefficients of Fourier-Vilenkin Series”, Positivity, 22:3 (2018), 897–918  crossref  mathscinet  isi  scopus
    12. L. S. Simonyan, “On convergence of the Fourier double series with respect to the Vilenkin systems”, Uch. zapiski EGU, ser. Fizika i Matematika, 52:1 (2018), 12–18  mathnet
    13. M. G. Grigoryan, “Ob absolyutnoi skhodimosti ryadov Fure–Khaara v metrike $L^p(0,1)$, $0<p<1$”, Issledovaniya po lineinym operatoram i teorii funktsii. 46, Zap. nauchn. sem. POMI, 467, POMI, SPb., 2018, 34–54  mathnet
    14. Grigoryan M.G. Sargsyan S.A., “Almost Everywhere Convergence of Greedy Algorithm With Respect to Vilenkin System”, J. Contemp. Math. Anal.-Armen. Aca., 53:6 (2018), 331–345  crossref  mathscinet  isi  scopus
  • Математический сборник Sbornik: Mathematics (from 1967)
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