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Fejér means of Vilenkin-Fourier series. (English) Zbl 1265.42099

Let \(\{m_k\}_{k=0}^\infty\) be a sequence of natural numbers that are greater than \(1\), let \(M_0=1\), \(M_{k+1}=M_km_k\), \(k\geq 1\), and let \(\mathbb Z_{m_k}\) be the \(m_k\)-th cyclic group with discrete topology and measure \(\mu(x)=1/m_k\) for all \(x\in \mathbb Z_{m_k}\). Then \(G_m:=\prod^\infty_{k=0}\mathbb Z_{m_k}\) is a compact Vilenkin group. The elements of \(G_m\) are represented by sequences \(x=(x_1,x_2,\dots)\), \(x_j\in\mathbb Z_{m_j}\), and \(I_n(x)=\{y\in G_m: y_0=x_0,\dots,y_{n-1}=x_{n-1}\}\). The \(\sigma\)-algebra generated by \(\{I_n(x):x\in G_m\}\) is denoted by \(F_n\). Let \(f=\{f^{(n)}\}_{n=0}^\infty\) be a martingale with respect to \(\{F_n\}^\infty_{n=0}\). Then \(f^*=\sup_{n\geq 0}| f^{(n)}| \) is the maximal function of \(f\). The Hardy martingale space \(H^p(G_m)\) consists of all martingales \(f\) such that \(\| f\| _{H^p}=\| f^*\| _{L^p(G_m)}<\infty\), \(0<p<\infty\). If \(\{\psi_i\}^\infty_{i=0}\) is the character system for \(G_m\), then \(\hat{f}(i):=\lim_{k\to\infty}\int_{G_m}f^{(k)}(x)\overline{\psi_i(x)}\,d\mu(x)\), where \(\mu\) is the normalized Haar measure on \(G_m\), and \(\sigma_n(f)=\sum\limits^{n-1}_{i=0}(1-i/n)\hat{f}(i)\chi_i\), \(n\geq 1\), \(\sigma^*f(x)=\sup\limits_{n\geq 1}| \sigma_n(f)(x)| \). The main result of the paper is the following.
There exists \(f\in H^{1/2}(G_m)\) such that \(\sup\limits_{n\geq 1}\| \sigma_n(f)\| _{1/2}=\infty\) and \(\| \sigma^*(f)\| _{1/2}=\infty\), where \(\| \cdot\| _{1/2}\) is the quasinorm in \(L^{1/2}(G_m)\).

MSC:

42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
60G42 Martingales with discrete parameter
60G46 Martingales and classical analysis
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