Summability with the speed of orthogonal series by the Euler-Knopp and Cesàro methods

Abstract

The series ∑Uk is said to be summable (Eλ,q) to u, if λn(En q−u)=o(1), where En q denotes the Euler-Knopp transform of the sequence of partial sums of the series, and λ={λn} is a positive increasing sequence. It is shown that all the methods (Eλ,q), q>0, are equivalent in the case of the orthogonal series ∑ckφk(x), φn∈Lμ 2[a,b], almost everywhere in the interval [a,b] if ∑λk 2Ck 2<∞ and λ belongs to the class ΛE, which is defined as ΛE={λ:λn(k+1)/(n+1)λk=O(1);k,n=0,1,⋯,k≤n}. A similar result for Cesàro summability (Cλ,α) is proved by replacing the En q-means by Cesàro means σn α and the class ΛE by the class Λc={λ:λn(k+1)τ/(n+1)τλk=O(1); k,n=0,1,⋯,k≤n,τ∈(0,1/2)}.