On (C,1) summability of integrable functions with respect to the Walsh-Kaczmarz system
Gát, G.
Studia Mathematica, Tome 129 (1998), p. 135-148 / Harvested from The Polish Digital Mathematics Library
Access to full text
Full (PDF)

Let G be the Walsh group. For f∈L1(G) we prove the a. e. convergence σf → f(n → ∞), where σn is the nth (C,1) mean of f with respect to the Walsh-Kaczmarz system. Define the maximal operator σ*f≔supn|σnf|. 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.

Publié le : 1998-01-01
EUDML-ID : urn:eudml:doc:216548 
@article{bwmeta1.element.bwnjournal-article-smv130i2p135bwm,
     author = {G. G\'at},
     title = {On (C,1) summability of integrable functions with respect to the Walsh-Kaczmarz system},
     journal = {Studia Mathematica},
     volume = {129},
     year = {1998},
     pages = {135-148},
     zbl = {0905.42016},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-smv130i2p135bwm}
}

Gát, G. On (C,1) summability of integrable functions with respect to the Walsh-Kaczmarz system. Studia Mathematica, Tome 129 (1998) pp. 135-148. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-smv130i2p135bwm/

[00000] [Bal] L. A. Balashov, Series with respect to the Walsh system with monotone coefficients, Sibirsk, Mat. Zh. 12 (1971), 25-39 (in Russian). | Zbl 0224.42010

[00001] [SCH1] F. Schipp, Certain rearrangements of series with respect the Walsh system, Mat. Zametki 18 (1975), 193-201 (in Russian).

[00002] [SCH2] F. Schipp, Pointwise convergence of expansions with respect to certain product systems, Anal. Math. 2 (1976), 63-75.

[00003] [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.

[00004] [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).

[00005] [SK1] V.A. Skvortsov, On Fourier series with respect to the Walsh-Kaczmarz system, ibid. 7 (1981), 141-150. | Zbl 0472.42014

[00006] [SK2] F. Schipp, L1 convergence of Fourier series with respect to the Walsh-Kaczmarz system, Vestnik Moskov. Univ. Ser. Mat. Mekh. 1981, no. 6, 3-6 (in Russian).

[00007] [WY] W. S. Young, On the a.e. convergence of Walsh-Kaczmarz-Fourier series, Proc. Amer. Math. Soc. 44 (1974), 353-358. | Zbl 0288.42005