Approximation of double Walsh–Fourier series by means of the matrix transform

István Blahota

doi:10.3842/umzh.v76i5.7397

István Blahota Institute of Mathematics and Computer Sciences, University of Nyíregyháza, Nyíregyháza, Hungary

DOI: https://doi.org/10.3842/umzh.v76i5.7397

Keywords: Walsh group; Walsh system; Walsh-Fourier series; N\

Abstract

UDC 517.5

We discuss the rate of approximation of partial sums of the double Walsh–Fourier series in the spaces $L^p(G^2),$ $1\leq p <\infty,$ and $C(G^2)$ by the matrix transform.

References

L. Baramidze, L.-E. Persson, G. Tephnadze, P. Wall, Sharp $H_{p}-L_{p}$ type inequalities of weighted maximal operators of Vilenkin–Nörlund means and its applications, J. Inequal. and Appl., Article 242 (2016). DOI: https://doi.org/10.1186/s13660-016-1182-1

I. Blahota, G. Gát, Norm and almost everywhere convergence of matrix transform means of Walsh–Fourier series, Acta Univ. Sapientiae Math., 15, № 2, 244–258 (2023).

I. Blahota, K. Nagy, Approximation by Θ-means of Walsh–Fourier series, Anal. Math., 44, № 1, 57–71 (2018).

I. Blahota, K. Nagy, Approximation by matrix transform of Vilenkin–Fourier series, Publ. Math. Debrecen, 99, № 1-2, 223–242 (2021). DOI: https://doi.org/10.5486/PMD.2021.9001

I. Blahota, K. Nagy, Approximation by Marcinkiewicz-type matrix transform of Vilenkin–Fourier series, Mediterr. J. Math., 19, Article 165 (2022). DOI: https://doi.org/10.1007/s00009-022-02105-3

I. Blahota, K. Nagy, M. Salim, Approximation by Θ-means of Walsh–Fourier series in dyadic Hardy spaces and dyadic homogeneous Banach spaces, Anal. Math., 47, 285–309 (2021). DOI: https://doi.org/10.1007/s10476-021-0083-9

I. Blahota, K. Nagy, G. Tephnadze, Approximation by Marcinkiewicz Θ-means of double Walsh–Fourier series, Math. Inequal. Appl., 22, № 3, 837–853 (2019).

I. Blahota, G. Tephnadze, A note on maximal operators of Vilenkin–Nörlund means, Acta Math. Acad. Paedagog. Nyházi., 32, № 2, 203–213 (2016).

Á. Chripkó, Weighted approximation via Θ-summations of Fourier–Jacobi series, Studia Sci. Math. Hungar., 47, № 2, 139–154 (2010).

T. Eisner, The Θ-summation on local fields, Ann. Univ. Sci. Budapest. Sect. Comput., 33, 137–160 (2011).

S. Fridli, P. Manchanda, A. H. Siddiqi, Approximation by Walsh–Nörlund means, Acta Sci. Math., 74, 593–608 (2008).

U. Goginava, On the approximation properties of Cesàro means of negative order of Walsh–Fourier series, J. Approx. Theory, 115, 9–20 (2002). DOI: https://doi.org/10.1006/jath.2001.3632

U. Goginava, K. Nagy, Matrix summability of Walsh–Fourier series, Mathematics, MDPI, 10, № 14 (2022). DOI: https://doi.org/10.3390/math10142458

E. Hewitt, K. Ross, Abstract harmonic analysis, vols. I, II, Springer-Verlag, Heidelberg (1963). DOI: https://doi.org/10.1007/978-3-662-40409-6

M. A. Jastrebova, On approximation of functions satisfying the Lipschitz condition by arithmetic means of their Walsh–Fourier series, Mat. Sb., 71, 214–226 (1966) (in Russian).

N. Memić, L.-E. Persson, G. Tephnadze, A note on the maximal operators of Vilenkin–Nörlund means with non-increasing coefficients, Stud. Sci. Math. Hungar., 53, № 4, 545–556 (2016).

F. Móricz, B. E. Rhoades, Approximation by weighted means of Walsh–Fourier series, Int. J. Math. Sci., 19, № 1, 1–8 (1996).

F. Móricz, A. Siddiqi, Approximation by Nörlund means of Walsh–Fourier series, J. Approx. Theory, 70, 375–389 (1992). DOI: https://doi.org/10.1016/0021-9045(92)90067-X

K. Nagy, Approximation by Nörlund means of quadratical partial sums of double Walsh–Fourier series, Anal. Math., 36, № 4, 299–319 (2010).

K. Nagy, Approximation by Nörlund means of double Walsh–Fourier series for Lipschitz functions, Math. Inequal. Appl., 15, № 2, 301–322 (2012).

F. Schipp, W. R. Wade, Norm convergence and summability of Fourier series with respect to certain product systems, Pure and Appl. Math. Approx. Theory, Marcel Dekker, New York etc. (1992), p. 437–452.

F. Schipp, W. R. Wade, P. Simon, J. Pál, Walsh series. An introduction to dyadic harmonic analysis, Adam Hilger, Bristol, New York (1990).

V. A. Skvortsov, Certain estimates of approximation of functions by Cesàro means of Walsh–Fourier series, Mat. Zametki, 29, 539–547 (1981) (in Russian). DOI: https://doi.org/10.1007/BF01343535

R. Toledo, On the boundedness of the $L^1$-norm of Walsh–Fejer kernels, J. Math. Anal. and Appl., 457, № 1, 153–178 (2018).

F. Weisz, $Theta$-summability of Fourier series, Acta Math. Hungar., 103, № 1-2, 139–175 (2004). DOI: https://doi.org/10.1023/B:AMHU.0000028241.87331.c5

F. Weisz, $Theta$-summation and Hardy spaces, J. Approx. Theory, 107, 121–142 (2000). DOI: https://doi.org/10.1006/jath.2000.3505

F. Weisz, Several dimensional $Theta$-summability and Hardy spaces, Math. Nachr., 230, 159–180 (2001). DOI: https://doi.org/10.1002/1522-2616(200110)230:1<159::AID-MANA159>3.0.CO;2-L

Sh. Yano, On Walsh–Fourier series, Tohoku Math. J., 3, 223–242 (1951). DOI: https://doi.org/10.2748/tmj/1178245527

Sh. Yano, On approximation by Walsh functions}, Proc. Amer. Math. Soc., 2, 962–967 (1951). DOI: https://doi.org/10.1090/S0002-9939-1951-0045235-4

A. Zygmund, Trigonometric series, 3rd ed., vols. 1, 2, Cambridge Univ. Press (2015).