# On (C,1) summability of integrable functions with respect to the Walsh-Kaczmarz system

Studia Mathematica (1998)

• Volume: 130, Issue: 2, page 135-148
• ISSN: 0039-3223

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## Abstract

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Let G be the Walsh group. For $f\in {L}^{1}\left(G\right)$ we prove the a. e. convergence σf → f(n → ∞), where ${\sigma }_{n}$ is the nth (C,1) mean of f with respect to the Walsh-Kaczmarz system. Define the maximal operator $\sigma *f≔su{p}_{n}|{\sigma }_{n}f|.$ We prove that σ* is of type (p,p) for all 1 < p ≤ ∞ and of weak type (1,1). Moreover, $\parallel \sigma *f{\parallel }_{1}\le c\parallel |f|{\parallel }_{H}$, where H is the Hardy space on the Walsh group.

## How to cite

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Gát, G.. "On (C,1) summability of integrable functions with respect to the Walsh-Kaczmarz system." Studia Mathematica 130.2 (1998): 135-148. <http://eudml.org/doc/216548>.

@article{Gát1998,
abstract = {Let G be the Walsh group. For $f ∈ L^1(G)$ we prove the a. e. convergence σf → f(n → ∞), where $σ_n$ is the nth (C,1) mean of f with respect to the Walsh-Kaczmarz system. Define the maximal operator $σ*f ≔ sup_n |σ_n f|.$ We prove that σ* is of type (p,p) for all 1 < p ≤ ∞ and of weak type (1,1). Moreover, $∥σ*f∥_1 ≤ c∥|f|∥_H$, where H is the Hardy space on the Walsh group.},
author = {Gát, G.},
journal = {Studia Mathematica},
keywords = {almost everywhere summability; Cesàro means; Walsh-Kaczmarz-Fourier series},
language = {eng},
number = {2},
pages = {135-148},
title = {On (C,1) summability of integrable functions with respect to the Walsh-Kaczmarz system},
url = {http://eudml.org/doc/216548},
volume = {130},
year = {1998},
}

TY - JOUR
AU - Gát, G.
TI - On (C,1) summability of integrable functions with respect to the Walsh-Kaczmarz system
JO - Studia Mathematica
PY - 1998
VL - 130
IS - 2
SP - 135
EP - 148
AB - Let G be the Walsh group. For $f ∈ L^1(G)$ we prove the a. e. convergence σf → f(n → ∞), where $σ_n$ is the nth (C,1) mean of f with respect to the Walsh-Kaczmarz system. Define the maximal operator $σ*f ≔ sup_n |σ_n f|.$ We prove that σ* is of type (p,p) for all 1 < p ≤ ∞ and of weak type (1,1). Moreover, $∥σ*f∥_1 ≤ c∥|f|∥_H$, where H is the Hardy space on the Walsh group.
LA - eng
KW - almost everywhere summability; Cesàro means; Walsh-Kaczmarz-Fourier series
UR - http://eudml.org/doc/216548
ER -

## References

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1. [Bal] L. A. Balashov, Series with respect to the Walsh system with monotone coefficients, Sibirsk, Mat. Zh. 12 (1971), 25-39 (in Russian). Zbl0224.42010
2. [SCH1] F. Schipp, Certain rearrangements of series with respect the Walsh system, Mat. Zametki 18 (1975), 193-201 (in Russian).
3. [SCH2] F. Schipp, Pointwise convergence of expansions with respect to certain product systems, Anal. Math. 2 (1976), 63-75.
4. [SWS] F. Schipp, W. R. Wade, P. Simon and J. Pál, Walsh Series: an Introduction to Dyadic Harmonic Analysis, Adam Higler, Bristol and New York, 1990.
5. [SN] A. A. Shneǐder, On series with respect to the Walsh functions with monotone coefficients, Izv. Akad. Nauk SSSR Ser. Mat. 12 (1948), 179-192 (in Russian).
6. [SK1] V.A. Skvortsov, On Fourier series with respect to the Walsh-Kaczmarz system, ibid. 7 (1981), 141-150. Zbl0472.42014
7. [SK2] F. Schipp, ${L}_{1}$ convergence of Fourier series with respect to the Walsh-Kaczmarz system, Vestnik Moskov. Univ. Ser. Mat. Mekh. 1981, no. 6, 3-6 (in Russian).
8. [WY] W. S. Young, On the a.e. convergence of Walsh-Kaczmarz-Fourier series, Proc. Amer. Math. Soc. 44 (1974), 353-358. Zbl0288.42005

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