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Cesàro means of integrable functions with respect to unbounded Vilenkin systems. (English) Zbl 1032.43003

The author proves a main theorem in this note: Theorem. Let \(f\in L^1(G_m)\). Then we have \(\sigma_{M_n}f\to f\), a.e., where \(m= (m_k: k\in \{0,1,2,\dots\})\) is a sequence of integers each of them \(\geq 2\). And \(Z_{m_k}\) denote the discrete cyclic group of order \(m_k\) with normalized Haar measure \(\mu_k\) on \(Z_{m_k}\); \(G_m= \prod^\infty_{k=0} Z_{m_k}\) is the unbounded Vilenkin group with infinite order \(\sup_n m_n= \infty\), and \(\sigma_nf\) is the Fejér mean.
To prove the theorem, the author introduces several operators, such as \(H_1f\), that is of \(s\)-\((2,2)\) type (proved in Lemma 2.3) and of \(w\)-\((1,1)\) type (proved in Lemma 2.4); \(H_jf\), \(j\geq 2\), which are of \(w\)-\((1,1)\) type (proved in Lemma 2.5). These operators play important roles in the proof of the main theorem.
Reviewer: Su Weiyi (Nanjing)

MSC:

43A70 Analysis on specific locally compact and other abelian groups
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
43A75 Harmonic analysis on specific compact groups
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