On the divergence of the (𝐶,1) means of double Walsh-Fourier series

Gát, G.

doi:10.1090/S0002-9939-99-05293-4

On the divergence of the $(C,1)$ means of double Walsh-Fourier series
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Proc. Amer. Math. Soc. 128 (2000), 1711-1720 Request permission

Abstract:

In 1992, Móricz, Schipp and Wade proved the a.e. convergence of the double $(C,1)$ means of the Walsh-Fourier series $\sigma _{n}f\to f$ ($\min (n_{1}, n_{2})\to \infty , n=(n_{1},n_{2})\in {\mathbb {N}} ^{2}$) for functions in $L\text {log}^{+} L(I^{2})$ ($I^{2}$ is the unit square). This paper aims to demonstrate the sharpness of this result. Namely, we prove that for all measurable function $\delta :[0,+\infty ) \to [0,+\infty ) , \lim _{t\to \infty }\delta (t)=0$ we have a function $f$ such as $f\in L\text {log}^{+} L\delta (L)$ and $\sigma _{n}f$ does not converge to $f$ a.e. (in the Pringsheim sense).

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