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\((C,\alpha)\) summability of Walsh-Fourier series. (English) Zbl 0992.42016

Define the martingale Hardy-Lorentz space \(H_{p,q}\) to be the set of dyadic martingales \(\{f_{n}\}_{n=1}^{\infty}\) for which \(\|\sup_{n\in\mathbb{N}}|f_{n}|\|_{p,q}<\infty\), where \(\|f\|_{p,q}\) is the \(L_{p,q}\)-norm of \(f\). This paper considers the maximal \((C,\alpha) \) means of the Walsh-Fourier series of \(f\in L_{1}([0,1)) \) \[ \sigma_{\ast}^{\alpha}(f) =\sup_{n\in\mathbb{N}}|\sigma_{n}^{\alpha}(f) | \] and shows that for \(0<\alpha\leq 1\), the operator is bounded from \(H_{p,q}\) to \(L_{p,q}\) for \(\frac{1}{1+\alpha}<p<\infty\) and \(0<q\leq\infty\), and maps \(L^{1}\) to \(L_{1,\infty}\). As a consequence of this result, the author derives norm and pointwise convergence results for the \((C,\alpha) \) means \(\{\sigma_{n}^{\alpha}f\}_{n=1}^{\infty}\), and also for the conjugate \((C,\alpha) \) means. This work generalizes a pointwise convergence result of N. J. Fine [Proc. Natl. Acad. Sci. USA 41, 558-591 (1955; Zbl 0065.05303)].

MSC:

42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
42A24 Summability and absolute summability of Fourier and trigonometric series
60G46 Martingales and classical analysis

Citations:

Zbl 0065.05303
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