A note on a theorem of Alexits and Králik

Abstract

where S , ( f ) is the n-th partial sum of (1). It is known (cf. ZYGMtYNO [4] Vol. I. p. 115) that [V.(f)-f l<=4e.(f) where E, ( f ) represents the best approximation of f(x) by trigonometric polynomials of order n. We denote by Z the class of all continuous 2n periodic functions satisfying the inequality sup [f(x +t ) -2 f (x )+f (xt ) t<= <-Kit[ for all x and t where K is an absolute positive constant. Earlier, G. AL~XITS and D. KgALIK [2] gave the necessary and sufficient condition for r-th derivative f(r)(x) of a function f(x) to belong to the class Z in terms of the de la ValiSe Poussin sum in strong sense: 1 2n--1 ~ I S k ( f ) f [ = O ( n , 1 ) . n k=n