Advertisement

Almost Everywhere Convergence of Greedy Algorithm with Respect to Vilenkin System

  • M. G. GrigoryanEmail author
  • S. A. Sargsyan
Real and Complex Analysis
  • 4 Downloads

Abstract

In this paper, we prove that for any ε ∈ (0, 1) there exists ameasurable set E ∈ [0, 1) with measure |E| > 1 − ε such that for any function fL1[0, 1), it is possible to construct a function \(\tilde f \in {L^1}[0,1]\) coinciding with f on E and satisfying \(\int_0^1 {|\tilde f(x) - f(x)|dx < \varepsilon } \), such that both the Fourier series and the greedy algorithm of \(\tilde f\) with respect to a bounded Vilenkin system are almost everywhere convergent on [0, 1).

Keywords

Vilenkin system convergence Fourier series greedy algorithm 

MSC2010 numbers

42C10 42C20 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    N.N. Luzin, “Sur le theore‘me fondamental du calcul integral”, Mat. Sbornik, 28 (2), 266–294, 1912.Google Scholar
  2. 2.
    D.E. Men’shov, “Sur la representation des fonctions measurables des series trigonometriques”, Mat. Sbornik, 9, 667–692, 1941.Google Scholar
  3. 3.
    A. A. Talalyan, “Dependence of convergence of orthogonal series on changes of values of the function expanded”, Mat. Zametki, 33 (5), 715–722, 1983.MathSciNetzbMATHGoogle Scholar
  4. 4.
    A. M. Olevskii, “Modifications of functions and Fourier series”, UMN, 40 (3), 157–193, 1985.MathSciNetGoogle Scholar
  5. 5.
    M.G. Grigorian, “On the Lp µ–strong property of orthonormal systems”, Matem. Sbornik, 194 (10), 1503–1532, 2003.CrossRefGoogle Scholar
  6. 6.
    M.G. Grigoryan, “On convergence of Fourier series in complete orthonormal systems in the L1 metric and almost everywhere”, Mat. Sb., 181 (8), 1011–1030, 1990.zbMATHGoogle Scholar
  7. 7.
    V.I. Golubov, A.F. Efimov, V.A. Skvorcov, Walsh series and transforms (Nauka, Noscow, 1987).Google Scholar
  8. 8.
    N. Ya. Vilenkin, “On a class of complete orthonormal systems”, Izv. AN SSSR, Ser. Mat., 11, 363–400, 1947.MathSciNetzbMATHGoogle Scholar
  9. 9.
    J.L. Walsh, “A closed set of normal orthogonal functions”, Amer. J.Math., 45, 5–24, 1923.MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    H.E. Chrestenson, “A class of generalizedWalsh functions”, Pacif. J.Math., 5 (1), 17–31, 1955.MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    C. Watari, “On generalizesWalsh–Fourier series”, IPProc. Japan Acad., 73 (8), 435–438, 1957.MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    W.S. Young, “Mean convergence of generalizedWalsh–Fourier series”, Trans. Amer.Math. Soc., 218, 311–320, 1976.MathSciNetCrossRefGoogle Scholar
  13. 13.
    J.J. Price, “Certain groups of orthonormal step functions”, Canad. J.Math., 9 (3), 413–425, 1957.MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    T. Ohkuma, “On a certain system of orthogonal step functions”, TohokuMath J., 5, 166–177, 1953.MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    P. Wojtaszczyk, “Greedy Algorithm for General Biorthogonal Systems”, Journal of Approximation Theory, 107, 293–314, 2000.MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    G.G. Gevorkyan, A. Kamont, “Two remarks on quasi–greedy bases in the space”, Journal of Contemporary Mathematical Analysis (National Academy of Sciences of Armenia), 40 (1), 2–14, 2005.MathSciNetzbMATHGoogle Scholar
  17. 17.
    T.W. Körner, “Divergence of decreasing rearranged Fourier series”, Ann. ofMath., 144, 167–180, 1996.MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    M.G. Grigoryan, S.A. Sargsyan, “Nonlinear approximation of functions of the class Lr by Vilenkin system”, Izv. vuz.,Matem., 2, 30–39, 2013.Google Scholar
  19. 19.
    M.G. Grigorian, R.E. Zink, “Greedy approximation with respect to certain subsystems of the Walsh orthonormal system”, Proc. of the Amer.Mat. Soc., 134 (12), 3495–3505, 2006.MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    M.G. Grigorian, “On the convergence in the metric of Lp of the greedy algorithm by triginometric system”, Journal of ContemporaryMathematical Analysis (National Academy of Sciences of Armenia), 39 (5), 37–52, 2004.Google Scholar
  21. 21.
    M.G. Grigorian, “Modifications of functions, Fourier coefficients and nonlinear approximation”, Mat. Sb., 203 (3), 49–78, 2012.MathSciNetCrossRefGoogle Scholar
  22. 22.
    M.G. Grigoryan, K.A. Navasardyan, “On behavior of Fourier coefficients by Walsh system”, Journal of ContemporaryMathematical Analysis (National Academy of Sciences of Armenia), 51 (1), 3–20, 2016.zbMATHGoogle Scholar
  23. 23.
    M.G. Grigorian, S.A. Sargsyan, “On the Fourier–Vilenkin coefficients”, ActaMathematica Scientia, 37(B) (2), 293–300, 2017.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Allerton Press, Inc. 2018

Authors and Affiliations

  1. 1.Yerevan State UmiversityYerevanArmenia

Personalised recommendations