Banach Journal of Mathematical Analysis

On the universal and strong (L1,L)-property related to Fourier–Walsh series

Martin G. Grigoryan

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text

Abstract

In this article, we construct a function UL1[0,1) with strictly decreasing Fourier–Walsh coefficients {ck(U)}, and having a universal and strong (L1,L)-property with respect to the Walsh system.

Article information

Source
Banach J. Math. Anal., Volume 11, Number 3 (2017), 698-712.

Dates
Received: 25 July 2016
Accepted: 12 October 2016
First available in Project Euclid: 16 May 2017

Permanent link to this document
https://projecteuclid.org/euclid.bjma/1494900292

Digital Object Identifier
doi:10.1215/17358787-2017-0012

Mathematical Reviews number (MathSciNet)
MR3679902

Zentralblatt MATH identifier
1376.42037

Subjects
Primary: 42A65: Completeness of sets of functions
Secondary: 42A20: Convergence and absolute convergence of Fourier and trigonometric series

Keywords
universal function Fourier–Walsh coefficients

Citation

Grigoryan, Martin G. 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. doi:10.1215/17358787-2017-0012. https://projecteuclid.org/euclid.bjma/1494900292


Export citation

References

  • [1] S. A. Episkoposian, On the existence of universal series by the generalized Walsh system, Banach J. Math. Anal. 10 (2016), no. 2, 415–429.
  • [2] S. A. Episkoposian and J. Müller, Universality properties of Walsh-Fourier series, Monatsh. Math. 175 (2014), no. 4, 511–518.
  • [3] G. G. Gevorkyan and A. Kamont, Two remarks on quasi-greedy bases in the space $L_{1}$, J. Contemp. Math. Anal. 40 (2005), no. 1, 2–14.
  • [4] B. Golubov, A. Efimov, and V. Skvortsov, Walsh Series and Transforms: Theory and Applications, Math. Appl. (Soviet Series) 64, Kluwer Academic, Dordrecht, 1991.
  • [5] M. G. Grigoryan, On the convergence of Fourier series in the metric of $L^{1}$, Anal. Math. 17 (1991), no. 3, 211–237.
  • [6] M. G. Grigoryan, Uniform convergence of the greedy algorithm with respect to the Walsh system, Studia Math. 198 (2010), no. 2, 197–206.
  • [7] M. G. Grigoryan, Modifications of functions, Fourier coefficients and nonlinear approximation (in Russian), Mat. Sb. 203 (2012), no. 3, 49–78; English translation in Sb. Math. 203 (2012), no. 3–4, 351–379.
  • [8] M. G. Grigoryan and L. N. Galoyan, On the uniform convergence of negative order Cesaro means of Fourier series, J. Math. Anal. Appl. 434 (2016), no. 1, 554–567.
  • [9] M. G. Grigoryan and A. A. Sargsyan, On the universal function for the class $L^{p}[0,1]$, $p\in(0,1)$, J. Funct. Anal. 270 (2016), no. 8, 3111–3133.
  • [10] A. Kamont, General Haar systems and greedy approximation, Studia Math. 145 (2001), no. 2, 165–184.
  • [11] S. V. Konyagin and V. N. Temlyakov, A remark on greedy approximation in Banach spaces, East J. Approx. 5 (1999), no. 3, 365–379.
  • [12] T. W. Körner, Decreasing rearranged Fourier series, J. Fourier Anal. Appl. 5 (1999), no. 1, 1–19.
  • [13] N. N. Luzin, On the fundamental theorem of the integral calculus (in Russian), Mat. Sb. 28 (1912), 266–294; English translation in Sb. Math. 28 (1912).
  • [14] D. E. Menchoff, Sur la convergence uniforme des series de Fourier, Sb. Math. 53 (1942), no. 11, 67–96.
  • [15] D. E. Menchoff, On Fourier series of summable functions, Trans. Moscow Math. Soc. 1 (1952), 5–38.
  • [16] K. A. Navasardyan, On series in the Walsh system with monotone coefficients (in Russian), Izv. Nats. Akad. Nauk Armenii Mat. 42 (2007), no. 5, 51–64; English translation in J. Contemp. Math. Anal. 42 (2007), no. 5, 258–269.
  • [17] A. M. Olevskii, Modifications of functions and Fourier series, Russian Math. Surveys 40 (1985), no. 3, 181–224.
  • [18] K. I. Oskolkov, The uniform modulus of continuity of summable functions on sets of positive measure (in Russian), Dokl. Akad. Nauk SSSR 229 (1976), no. 2, 304–306; English translation in Dokl. Math. 229 (1976), no. 2, 1028–1030.
  • [19] R. E. A. C. Paley, A remarkable set of orthogonal functions, I, Proc. Lond. Math. Soc. 34 (1932), 241–264; II, 265–279.
  • [20] J. J. Price, Walsh series and adjustment of functions on small sets, Illinois J. Math. 13 (1969), 131–136.
  • [21] J. L. Walsh, A closed set of normal orthogonal functions, Amer. J. Math. 45 (1923), no. 1, 5–24.
  • [22] P. Wojtaszczyk, Greedy algorithm for general biorthogonal systems, J. Approx. Theory 107 (2000), no. 2, 293–314.