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)


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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