Advances in Operator Theory

On behavior of Fourier coefficients and uniform convergence of Fourier series in the Haar system

M. G. ‎Grigoryan‎ and A. Kh. ‎Kobelyan

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Abstract

‎Suppose that $\hat{b}_m\downarrow 0,\ \{\hat{b}_m\}_{m=1}^\infty\notin l^2,$‎ ‎and $b_n=2^{-\frac{m}{2}}\hat{b}_m$ for all $ n\in(2^m,2^{m+1}].$‎ ‎In this paper‎, ‎it is proved that any measurable and almost everywhere finite‎ ‎function $f(x)$ on $[0,1]$ can be corrected on a set of arbitrarily small measure‎ ‎to a bounded measurable function $\widetilde{f}(x)$; so that the nonzero Fourier-Haar‎ ‎coefficients of the corrected function present some subsequence of $\{b_n\}$‎, ‎and its‎ ‎Fourier-Haar series converges uniformly on $[0,1]$‎.

Article information

Source
Adv. Oper. Theory, Volume 3, Number 4 (2018), 781-793.

Dates
Received: 21 January 2018
Accepted: 12 May 2018
First available in Project Euclid: 8 June 2018

Permanent link to this document
https://projecteuclid.org/euclid.aot/1528444822

Digital Object Identifier
doi:10.15352/aot.1801-1300

Mathematical Reviews number (MathSciNet)
MR3856172

Subjects
Primary: 42A65: Completeness of sets of functions
Secondary: 42A20‎ ‎43A50

Keywords
Haar system ‎Fourier-Haar coefficients ‎Uniform convergence

Citation

‎Grigoryan‎, M. G.; ‎Kobelyan, A. Kh. On behavior of Fourier coefficients and uniform convergence of Fourier series in the Haar system. Adv. Oper. Theory 3 (2018), no. 4, 781--793. doi:10.15352/aot.1801-1300. https://projecteuclid.org/euclid.aot/1528444822


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References

  • F. G. Arutyunyan, On series in the Haar system, ANA SSR Dokl. 42 (1966), no.3, 134–140.
  • R. A. DeVore, V. N. Temlyakov, Some remarks on greedy algorithms, Adv. Comput. Math. 5 (1966) no. 2-3, 173–187.
  • S. A. Episkoposyan, On the existence of universal series by the generalized Walsh system , Banach J. Math. Anal. 10 (2016), no. 2, 415–429.
  • M. G. Grigorian, On the convergence of Fourier series in the metric of $L^{1}$, Anal. Math. 17 (1991), 211–237.
  • M. G. Grigoryan, On the universal and strong ($L^1$,$L^{\infty }$)-property related to Fourier–Walsh series, Banach J. Math. Anal. 11 (2017), no. 3, 698–712.
  • M. G. Grigoryan, V. G. Krotov, Luzin's Correction Theorem and the Coefficients of Fourier Expansions in the Faber-Schauder System, Math. Notes 93 (2013), no. 2, 11–17.
  • A. Haar, Zur Theorie der orthogonalen Funktionensysteme , Math. Ann. 69 (1910), no. 3, 331–371.
  • N. N. Luzin, On the basic theorem of integral calculus, Mat. Sbornik 28 (1912), 266–294.
  • J. Marcinkiewicz, Quelques théoremes sur les séries orthogonales, Ann. Soc. Polon. Math., 16 (1937), 84–96 (pp. 307–318 of the Collected Papers).
  • D. E. Menchoff, Sur la convergence uniforme des séries de Fourier , Mat. Sbornik 53 (1942), no. 1-2, 67–96.(French)
  • K. A. Navasardyan, A. A. Stepanyan, Series by Haar system, Izv. Nats. Akad. Nauk Armenii mat. 42 (2007), no. 4, 53–66.
  • A. M. Olevskii, Modification of functions and Fourier series, Uspekhi Mat. Nauk, 40 (1985), no. 3(243), 157–193.
  • K. I. Oskolkov, The uniform modulus of continuity of integrable functions on sets of positive measure, Dokl. Akad. Nauk SSSR 229 (1976), 304–306.
  • J. J. Price, Walsh series and adjustment of functions on small sets, Illinois J. Math. 13 (1969), no. 1,131–136.