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Zbl 0905.42016
Gát, G.
On $(C,1)$ summability of integrable functions with respect to the Walsh-Kaczmarz system. (English)
[J] Stud. Math. 130, No.2, 135-148 (1998). ISSN 0039-3223; ISSN 1730-6337/e

{\it W.-S. Young} [Proc. Am. Math. Soc. 44, 353-358 (1974; Zbl 0288.42005)] proved that under the Kaczmarz rearrangement, the Walsh-Fourier series of a function in $L^1(\log^+L)^2$ converges almost everywhere on $[0,1]$. In this paper, the author proves that under the Kaczmarz rearrangement, the Walsh-Fourier series of a function in $L^1$ is Cesàro summable almost everywhere. The method of proof uses Schipp's decomposition of the corresponding Fejér kernel to show the associated maximal function $\sigma^*$ is of weak type $(1,1)$. In the process, the author also shows that $\sigma^*$ is of type $(H^1, L^1)$.
[W.R.Wade (Knoxville)]

MSC 2000:: *42C10 Fourier series in special orthogonal functions
40G05 Traditional summability methods

Keywords: almost everywhere summability; Cesàro means; Walsh-Kaczmarz-Fourier series

Citations: Zbl 0288.42005

Cited in: Zbl 0987.42021

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