Fourier – Vilenkin series

Λ-Summability and Multiplicators of Holder Classes of Fourierseries with Respect ̈ to Character Systems

Authors: Agafonova N. Y.

Let G be a Vilenkin group of bounded type. We obtain nessesary and sufficient conditions of uniform Λ-summability for all Fourier series of f ∈ C(G) and one of Λ-summability in L 1 (G) for all Fourier series of f ∈ L 1 (G). Also we extend some T. Quek and L. Yap results to the case of general modulus of continuity.

Absolute Convergence of Single and Double Fourier Series on Multiplicative Systems

Authors: Volosivets S. S.

Two-dimensional analogs of famous Zygmund and Szasz tests for absolute convergence of Fourier – Vilenkin series are established. Also it is proved that two-dimensional Szasz test is the best possible in the certain sense.

On Uniform Convergenceof Transformationsof Fourier Serieson Multiplicative Systems

Authors: Agafonova N. Y.

Necessary and suffiecient conditions for uniform Λ-summability of Fourier – Vilenkin series of Functions from Orlicz spaces LΦ[0,1) and L1[0,1) are obtained. Some corollaries for matrices with generalized monotone coeffiecients are given.

On the L1-convergence of Series in Multiplicative Systems

Authors: Agafonova N. Y.

In the paper two analogs of Garrett – Stanojevic´ trigonometric results are established for multiplicative systems {χn} ∞n=0 of bounded type. First, the modified partial sums of a series P∞ k=0 akχk with coefficients of bounded variation converge in L 1 [0, 1) to its sum if and only if for all ε > 0 there exists δ > 0 such that R δ 0 ¯ ¯ ¯ ¯ P∞ k=n (ak − ak+1)Dk+1(x) ¯ ¯ ¯ ¯ dx < ε, n ∈ Z+, where Dk+1(x) = Pk i=0 χi(x).