×

Approximation by Nörlund means of quadratical partial sums of double Walsh-Fourier series. (English) Zbl 1240.42133

The main theorem of the paper states that the approximation behavior of the two-dimensional Walsh-Nörlund means of Marcinkiewicz type is so good as the approximation behavior of the one-dimensional Walsh-Nörlund means. More precisely, let \[ \boldsymbol{t}^w_n(f, x^1,x^2) :=\frac{1}{Q_n}\sum_{k=1}^nq_{n-k}S_{k,k}^w(f, x^1,x^2) \] be the two dimensional Walsh-Nörlund means of Marcinkiewicz type, where \(Q_n=\sum_{k=1}^{n-1}q_k\).
Then if \(f\in L^p\) (\(1\leq p\leq\infty\)) and \(q_k\) is nondecreasing, we have \[ \| \boldsymbol{t}_n^w(f)-f\| _p\leq\frac{c}{Q_n}\sum_{l=0}^{A-1}q_{n-2^l}2^l\omega_p(2^{-l},f)+O(\omega_p(2^{-A},f)), \] and the same for nonincreasing \(q_k\) if we additionally suppose that \[ \frac{n}{Q_n^2}\sum_{k=1}^{n-1}q_k^2=O(1). \] Here, \(A\) is the order of \(n\), supposed to be positive, and \(\omega_p(\delta,f)\) is the modulus of continuity.
The author discusses also the asymptotic behaviour of the norm of \(\boldsymbol{t}_n^w(f)-f\) if \(f\) is in different Lipschitz classes.

MSC:

42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] G. H. Agaev, N. Ja. Vilenkin, G. M. Dzhafarli, and A. I. Rubinstein, Multiplicative systems of functions and harmonic analysis on 0-dimensional groups, Izd. ”ELM” (Baku, 1981) (in Russian).
[2] I. Blahota and G. Gát, Norm summability of Nörlund logarithmic means on unbounded Vilenkin groups, Anal. Theory Appl., 24(1)(2008), 1–17. · Zbl 1164.42022 · doi:10.1007/s10496-008-0001-z
[3] S. Fridli, P. Manchanda, and A. H. Siddiqi, Approximation by Walsh-Nörlund means, Acta Sci. Math. (Szeged), 74(2008), 593–608. · Zbl 1199.42124
[4] G. Gát and U. Goginava, Uniform and L-convergence of logarithmic means of Walsh-Fourier series, Acta Math. Sin., (Eng. Ser.) 22(2)(2006), 497–506. · Zbl 1129.42411 · doi:10.1007/s10114-005-0648-8
[5] G. Gát and U. Goginava, Uniform and L-convergence of logarithmic means of cubical partial sums of double Walsh-Fourier series, East J. Approx., 10(3)(2004), 391–412. · Zbl 1113.42006
[6] G. Gát and R. Toledo, L p-norm convergence of series in compact, totally disconnected groups, Analysis Math., 22(1996), 13–24. · Zbl 0856.42017 · doi:10.1007/BF02342335
[7] V. A. Glukhov, On the summability of multiple Fourier series with respect to multiplicative systems Mat. Zametki, 39(1986), 665–673 (in Russian).
[8] U. Goginava and G. Tkebuchava, Convergence of subsequences of partial sums and logarithmic means of Walsh-Fourier series, Acta Sci. Math. (Szeged), 72(2006), 159–177. · Zbl 1109.42008
[9] U. Goginava, On the approximation properties of Cesàro means of negative order of Walsh-Fourier series, J. Approx. Theory, 115(2002), 9–20. · Zbl 0998.42018 · doi:10.1006/jath.2001.3632
[10] U. Goginava, Approximation properties of (C, {\(\alpha\)}) means of double Walsh-Fourier series, Anal. Theory Appl., 20(1)(2004), 77–98. · Zbl 1078.42020 · doi:10.1007/BF02835261
[11] M. A. Jastrebova, On approximation of functions satisfying the Lipschitz condition by arithmetic means of their Walsh-Fourier series, Math. Sb., 71(1966), 214–226 (in Russian). · Zbl 0191.36802
[12] F. Móricz and B. E. Rhoades, Approximation by Nörlund means of double Fourier series for Lipschitz functions, J. Approx. Theory, 50(1987), 341–358. · Zbl 0636.42003 · doi:10.1016/0021-9045(87)90012-8
[13] F. Móricz and F. Schipp, On the integrability and L 1 convergence of Walsh series with coefficients of bounded variation, J. Math. Anal. Appl., 146(1)(1990), 99–109. · Zbl 0693.42023 · doi:10.1016/0022-247X(90)90335-D
[14] F. Móricz and A. Siddiqi, Approximation by Nörlund means of Walsh-Fourier series, J. Approx. Theory, 70(3)(1992), 375–389. · Zbl 0757.42009 · doi:10.1016/0021-9045(92)90067-X
[15] F. Schipp, W. R. Wade, P. Simon, and J. Pál, Walsh Series. An Introduction to Dyadic Harmonic Analysis, Adam Hilger (Bristol-New York, 1990). · Zbl 0727.42017
[16] V.A. Skvortsov, Certain estimates of approximation of functions by Cesàro means of Walsh-Fourier series, Mat. Zametki, 29(1981), 539–547 (in Russian). · Zbl 0467.42021
[17] Sh. Yano, On Walsh series, Tôhoku Math. J., 3(1951), 223–242. · Zbl 0044.07101 · doi:10.2748/tmj/1178245527
[18] Sh. Yano, On approximation by Walsh functions, Proc. Amer. Math. Soc., 2(1951), 962–967. · Zbl 0044.07102 · doi:10.1090/S0002-9939-1951-0045235-4
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.