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Norm convergence of logarithmic means on unbounded Vilenkin groups. (English) Zbl 1394.42019
It is proved that if a sequence \((m_n)\) satisfies the condition \[ \sup_{n\in\mathbb{N}}\;\frac{\ln^2 m_1+\dots+\ln^2 m_n}{\ln m_1+\dots+\ln m_n} <\infty, \] then for the Vilenkin system generated by the sequence \((m_n)\) the convergence is guaranteed of the Riesz logarithmic means in the spaces \(L\) and \(C\). Consequently, an example is found of an unbounded Vilenkin system for which there exists a linear summation method providing the summability of the Fourier-Vilenkin series in the spaces \(L\) and \(C\).

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