Behavior of Lebesgue constants for linear methods of summing fourier series giving the best order approximation

Abstract

co ~6R. We assume that the series ~ (I/~(k))Ak(f,x+~/2k) is the Fourier series of some function df k = l U ~(')6 ~, whereU~----{~:~EL ,[I~[]~.~I}.. We will denote the set of all continuous functions f which satisfy these conditions by C~,~. In addition, we assume that ~6F (~, i.e., that satisfies the following conditions: i) ~(v) is convex downwards for all v ~ 0, and |im t~(v)= di n ~ o o 0; 2)]~'(%t)l~K, t~t0>0 , where~(~;,h=~-~(~;(t)/2), t>0, and ~-i(.) is the inverse of ~('). Also, we will say that ~(') belongs to the set P if A~(I/~(k))= I/~(k)--2/~2(k--l)+ I/~(k--2)>~0.