Approximation by Vilenkin-Fourier sums
The author deals with the connection among the Vilenkin-Fourier sums, the modulus of continuity and the Lebesgue-constants with respect to the Vilenkin-system. The author proves five interesting theorems and he remarks that three of them can be also proved in the classical spaces \(C(\ell)\) and \(L^ p(\ell)\) \((\ell \leq p<\infty)\) with the classical modulus of continuity. For notations we refer to the paper. Six theorems are proved. We give the first as an example: 1. For all sequences \(m:=\{m_ k(\geq 2)\}\) and \(f\in L^ p(G_ m)(1<p<\infty)\liminf_{k\to \infty}\| f-S_ kf\|_ p/\omega (f,L^ p,1/k)L_ k=0.\)
scientific article; zbMATH DE number 4036408