A few remarks on Riesz summability of orthogonal series

Abstract

We study convergence behavior of some sequences and series related to a given orthogonal series. Following the developed technique we define (in terms of fourth mixed moments only) a class of orthonormal functions { X i } i ≥ 1 {\left \{ {{X_i}} \right \}_{i \geq 1}} such that the condition: ∃ k ∈ N ∑ i ≥ 1 μ i 2 ( ln ( k ) i ) 2 > ∞ \exists k \in \mathbb {N}\sum \nolimits _{i \geq 1} {\mu _i^2} {\left ( {{{\ln }^{\left ( k \right )}}i} \right )^2} > \infty implies almost everywhere convergence of the series ∑ i ≥ 1 μ i X i \sum \nolimits _{i \geq 1} {{\mu _i}{X_i}} , here for every i = 1 , 2 , … , j = 1 , … , k i = 1,2, \ldots ,j = 1, \ldots ,k , \[ ln ( 1 ) i = ln 2 i , ln ( j ) i = ln 2 ( max ( 1 , ln ( j − 1 ) i ) ) . {\ln ^{(1)}}i = {\ln _2}i,\quad {\ln ^{(j)}}i = {\ln _2}(\max (1,{\ln ^{(j - 1)}}i)). \]