## Banach Journal of Mathematical Analysis

### On the universal and strong $(L^{1},L^{\infty})$-property related to Fourier–Walsh series

Martin G. Grigoryan

#### Abstract

In this article, we construct a function $U\in L^{1}[0,1)$ with strictly decreasing Fourier–Walsh coefficients $\{c_{k}(U)\}\searrow$, and having a universal and strong $(L^{1},L^{\infty})$-property with respect to the Walsh system.

#### Article information

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

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

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

#### 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

#### 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.