## Tohoku Mathematical Journal

### Approximation by Cesàro means of negative order of double Walsh-Kaczmarz-Fourier series

Károly Nagy

#### Abstract

In this article we investigate the rate of the approximation by Cesàro means of the quadratical partial sums of double Walsh-Kaczmarz-Fourier series of a function in the Lebesgue space over the Walsh group. The approximation properties of Cesàro means of negative order of one- and two-dimensional Walsh-Fourier series was discussed earlier by Goginava.

#### Article information

Source
Tohoku Math. J. (2), Volume 64, Number 3 (2012), 317-331.

Dates
First available in Project Euclid: 11 September 2012

https://projecteuclid.org/euclid.tmj/1347369366

Digital Object Identifier
doi:10.2748/tmj/1347369366

Zentralblatt MATH identifier
1256.42042

#### Citation

Nagy, Károly. Approximation by Cesàro means of negative order of double Walsh-Kaczmarz-Fourier series. Tohoku Math. J. (2) 64 (2012), no. 3, 317--331. doi:10.2748/tmj/1347369366. https://projecteuclid.org/euclid.tmj/1347369366

#### References

• G. H. Agaev, N. Ja. Vilenkin, G. M. Dzhafarli and A. I. Rubinstein, Multiplicative systems of functions and harmonic analysis on zero-dimensional groups, (in Russian), Izd. (“Elm"), Baku, 1981.
• G. Gát, On $(C,1)$ summability of integrable functions with respect to the Walsh-Kaczmarz system, Studia Math. 130 (2) (1998), 135–148.
• V. A. Glukhov, Summation of multiple Fourier series in multiplicative systems, (in Russian), Mat. Zametki 39 (1986), 665–673.
• U. Goginava, On the convergence and summability of $N$-dimensional Fourier series with respect to the Walsh-Paley systems in the spaces $L^p([0,1]^N), p\in [1,\infty]$, Georgian Math. J. 7 (2000) 53–72.
• U. Goginava, On the approximation properties of Cesàro means of negative order of Walsh-Fourier series, J. Approx. Theory 115 (2002), 9–20.
• U. Goginava, Approximation properties of ($C,\alpha$) means of double Walsh-Fourier series, Anal. Theory Appl. 20 (1) (2004), 77–98.
• U. Goginava, Uniform convergence of Cesàro means of negative order of double trigonometric Fourier series, Anal. Theory Appl. 23 (3) (2007), 255–265.
• B. I. Golubov, A. V. Efimov and V. A. Skvortsov, Walsh series and transformations, (in Russian), Nauka, Moscow, Math. Appl. (Soviet Ser.) 64, 1987, English transl.: Kluwer Acad. publ., Dordrecht, 1991.
• F. Schipp, W. R. Wade, P. Simon and J. Pál, Walsh series. An introduction to dyadic harmonic analysis, Adam Hilger, Bristol-New York, 1990.
• F. Schipp, Pointwise convergence of expansions with respect to certain product systems, Anal. Math. 2 (1976), 65–76.
• P. Simon, (C,$\alpha$) summability of Walsh-Kaczmarz-Fourier series, J. Approx. Theory 127 (2004), 39–60.
• V. A. Skvortsov, On Fourier series with respect to the Walsh-Kaczmarz system, Anal. Math. 7 (1981), 141–150.
• A. A. Šneider, On series with respect to the Walsh functions with monotone coefficients, Izvestiya Akad. Nauk SSSR Ser. Mat. 12 (1948), 179–192.
• V. I. Tevzadze, Uniform convergence of Cesàro means of negative order of Walsh-Fourier series, (in Russian), Soobshch. Acad. Nauk. Gruzii SSR 102 (1981), 33–36.
• V. I. Tevzadze, Uniform $(C,\alpha)$ $(-1<\alpha<0)$ summability of Fourier series with respect to the Walsh-Paley system, Acta Math. Acad. Paedagog. Nyházi. (N.S.) 22 (2006), 41–61.
• W. S. Young, On the a.e. convergence of Walsh-Kaczmarz-Fourier series, Proc. Amer. Math. Soc. 44 (1974), 353–358.
• L. V. Zhizhiashvili, Trigonometric Fourier series and their conjugates, (in Russian), Tbilisi, 1993; English transl.: Kluwer Acad. publ., Dordrecht, 1996.
• A. Zygmund, Trigonometric series, Cambridge Univ. Press, Cambridge, UK, 1959.