Convergence in L p -norm of Fourier series on the complete product of quaternion groups with bounded orders

A well-known result for Vilenkin systems is the fact that for all 1 p ∞ the n-th partial sums of Fourier series of all functions in the space Lpconverge to the function in Lp-norm.This statement can not be generalized to any representative product system on the complete product of finite non-abelian groups,but even then it is true for the complete product of quaternion groups with bounded orders and monomial representative product system ordered in a specific way.     

Almost everywhere convergence of Cesàro means of Fourier series on the group of 2-adic integers

We prove the almost everywhere convergence of the Cesàro (C, α)-means of integrable functions σ _n ^α f → f for f ∈ L ¹(I), where I is the group of 2-adic integers for every α > 0. This theorem for the case of α = 1 was proved by the author [1]. For the case of the (C, 1) Fejér means there are several generalizations known with respect to some orthonormal systems. One could mention the papers [2, 9]. Research supported by the Hungarian National Foundation for Scientific Research (OTKA), grant no. T 048780.     

Sharp estimates for the convergence rate of Fourier series of complex variable functions in <Emphasis Type="Italic">L</Emphasis><Subscript>2</Subscript>(D, <Emphasis Type="Italic">p</Emphasis>(<Emphasis Type="Italic">z</Emphasis>))

Sharp estimates are derived for the convergence rate of Fourier series in terms of orthogonal systems of functions for certain classes of complex variable functions, and the Kolmogorov N-widths of these classes are determined. These issues find applications in numerical analysis methods.