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Gát, György; Goginava, Ushangi

Norm convergence of logarithmic means on unbounded Vilenkin groups. (English) Zbl 1394.42019

Banach J. Math. Anal. 12, No. 2, 422-438 (2018).
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 \(C\).
Reviewer: Giorgi Oniani (Kutaisi)

Cited in 6 Documents

MSC:

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

Keywords:

unbounded Vilenkin group; Riesz logarithmic means; convergence in \(L\) norm; summation
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\textit{G. Gát} and \textit{U. Goginava}, Banach J. Math. Anal. 12, No. 2, 422--438 (2018; Zbl 1394.42019)
Full Text: DOI Euclid

References:

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