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Strong convergence theorem for Vilenkin-Fourier series. (English) Zbl 0987.42022

Let \(m=(m_0,m_1,...,m_k,...)\) be a sequence of natural numbers such that \(m_k\geq 2, G_m\) be the complete direct product of \(Z_{m_k}\), and \(\|f\|_p= (\int_{G_m}|f|^pd\mu)^{1/p}, d\mu\) be the Haar measure on \(G_m\). Let \((\psi_n)_{n=0}^\infty\) be the character system on \(G_m\), \(S_n(f)\) be an \(n\)-th partial sum of \(f \) with respect to \((\psi_n)^\infty_{n=0}\). If \(f^\ast\) is the maximal martingale function of \(f\), then the Hardy space \(H^p\) with \(p>0\) consists of all \(f\) with \(\|f\|_{H^p}= \|f^\ast\|_p < \infty\). Theorem. Let \(1/2<p<1\) and let \(m\) be arbitrary. Then there exists a constant \(C_p\) such that for all \(f\in H^p\) \[ \sum^\infty_{k=1} \|S_k(f)\|^p_pk^{p-2}\leq C_p\|f\|^p_{H^p}. \] If \(m\) is a bounded sequence then the inequality holds for every \(0<p<1\).

MSC:

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