Real Analysis Exchange

OntheFourier-Walsh Coefficients

Martin G. Grigoryan

Full-text: Open access

Abstract

For any $0<\epsilon <1,\ p\geq 1$ and each function $f\in L^{p}[0,1]$ one can find a function $g\in L^{p}[0,1],\ mes\{x\in \lbrack 0,1] ;\ g\neq f\}<\epsilon $, such that the sequence $\{|c_{k}(g)|,\ k\in spec(g)\}$ is monotonically decreasing, where $\{c_{k}(g)\}$ is\ the sequence of Fourier-Walsh coefficients of the function $g(x)$.

Article information

Source
Real Anal. Exchange, Volume 35, Number 1 (2009), 157-166.

Dates
First available in Project Euclid: 27 April 2010

Permanent link to this document
https://projecteuclid.org/euclid.rae/1272376229

Mathematical Reviews number (MathSciNet)
MR2657293

Subjects
Primary: 42C10: Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.) 42C20: Other transformations of harmonic type
Secondary: 26D15: Inequalities for sums, series and integrals

Keywords
Fourier coefficients orthonormal system functional series

Citation

Grigoryan, Martin G. OntheFourier-Walsh Coefficients. Real Anal. Exchange 35 (2009), no. 1, 157--166. https://projecteuclid.org/euclid.rae/1272376229


Export citation

References

  • N. N. Luzin, On the fundamental theorem of the integral calculus, Mat. Sbornik, 28 (1912), 266–294 (in Russian).
  • D. E. Men'shov, Sur la representation des fonctions measurables des series trigonometriques, Mat. Sbornik, 9 (1941), 667–692.
  • D. E. Men'shov, On Fourier series of integrable functions, Trudy Moskov. Mat. Obshch., 1 (1952), 5–38.
  • F. G. Arutjunjan, Series with respect to the Haar system, (Russian. Armenian summary), Akad. Nauk Armjan. SSR Dokl., 42(3) (1966), 134–140.
  • J. J. Price, Walsh series and adjustment of functions on small sets, Illinois J. Math., 13 (1969), 131–136.
  • A. M. Olevskii, Modifications of functions and Fourier series. (Russian) Uspekhi Mat. Nauk 40 (1985), 157–193.
  • K. I. Oskolkov, The uniform modulus of continuity of summable functions on sets of positive measure, (Russian), Dokl. Akad. Nauk SSSR, 229(2) (1976), 304–306.
  • M. G. Grigoryan, Convergence in the $L\sp 1$-metric and almost everywhere convergence of Fourier series in complete orthonormal systems, (Russian), Mat. Sb., 181(8) (1990), 1011–1030; translation in Math. USSR-Sb., 70(2) (1991), 445–466.
  • M. G. Grigoryan, On the representation of functions by orthogonal series in weighted $L^p$ spaces, Studia. Math., 134(3) (1999), 207–216.
  • M. G. Grigoryan, On the strong $L^p_\mu$-property of orthonormal systems, (Russian. Russian summary), Mat. Sb., 194(10) (2003), 77–106; translation in Sb. Math., 194(9-10) (2003), 1503–1532.
  • B. I. Golubov, A. F. Efimov, V. A. Skvortsov, (Russian), Walsh Series and Transforms, Theory and Applications, “Nauka”, Moscow, 1987.
  • R. E. A. C. Paley, A remarkable set of orthogonal functions, Proc. London Math. Soc., 34 (1932), 241–279.
  • R. A. DeVore, V. N. Temlyakov, Some remarks on greedy algorithms, Advances in Computational Math., 5 (1996), 173–187.
  • S. V. Konyagin and V. N. Temlyakov, A remark on Greedy approximation in Banach spaces, East Journal on Approx., 5(1) (1999), 1–15.
  • V. N. Temlyakov, Nonlinear methods of approximation, Found. Comput. Math., 3 (2003), 33–107.
  • P. Wojtaszczyk, Greedy algorithm for general biorthogonal systems, Journal of Approx. Theory, 107 (2000), 293–314.
  • R. Gribonval, M. Nielsen, On the quasi-greedy property and uniformly bounded orthonormal systems, http://www.math.auc.dk/research/reports/R-2003-09.pdf.
  • T. W. K$\ddot{o}$rner, Divergence of decreasing rearranged Fourier series, Ann. of Math., 144 (1996), 167–180.
  • M. G. Grigoryan and R. E. Zink, Greedy approximation with respect to certain subsystems of the Walsh orthonormal system, Proc. of the Amer. Mat. Soc., 134(12) (2006), 3495–3505.
  • M. G. Grigoryan, K. S. Kazarian, F. Soria, Mean convergence of orthogonal Fourier series of mod. func., Trans. Amer. Math. Soc., 352(8) (2000), 3777–3798.