arXiv:2610.35900v1 Announce Type: new Abstract: We introduce a new logarithmic digit variation for Vilenkin--Fourier partial sums and establish two-sided estimates for the corresponding Lebesgue constants with universal constants. These estimates hold for both bounded and unbounded Vilenkin systems. As an application, we show that bounded logarithmic variation yields uniform boundedness of the corresponding partial-sum operators on \(L_1\), \(L_1\)-norm convergence for every integrable function, and almost everywhere convergence.
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