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Strong convergence theorems for Walsh-Fejér means. (English) Zbl 1313.42086

F. Weisz proved in Chapter 3 of [Summability of multi-dimensional Fourier series and Hardy spaces, Kluwer Academic, Dordrecht (2002; Zbl 1306.42003)] a series of results about \( (C,\alpha) \) summability of \(d\)-dimensional Walsh-Fourier series in Hardy spaces \( H_p\) with some restrictions on \(p\). In particular, for \(d=1\) and \(\alpha=1\) (Walsh-Fejér means \(\sigma_k)\) it gives the boundedness of \(\sigma_k\) in \(H_p\) spaces with \(p>1/2\) and as a consequence a strong convergence estimate for \(\sigma_k\).
Here the author proves the strong convergence estimate \[ \frac{1}{\log ^{[1/2+p]}n}\sum_{k=1}^{n} \frac{\| \sigma_k F\| _{H_p}^{p}}{k^{2-2p}}\leq c_p\| F\| _{H_p}^{p} \] for \( 0<p\leq 1/2\).

MSC:

42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)

Citations:

Zbl 1306.42003
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References:

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