Concerning the order approximation of periodic continuous functions by trigonometric interpolation polynomials II

Abstract

Letxkn=2θk/n,k=0,1 …n−1 (n odd positive integer). LetRn(x) be the unique trigonometric polynomial of order 2n satisfying the interpolatory conditions:Rn(xkn)=f(xkn),Rn(j)(xkn)=0,j=1,2,4,k=0,1…,n−1. We setw2(t,f) as the second modulus of continuity off(x). Then we prove that |Rn(x)-f(x)|=0(nw2(1/nf)). We also examine the question of lower estimate of ‖Rn-f‖. This generalizes an earlier work of the author.