Some approximation properties of hexagonal fourier series

Abstract

L. Leindler, A. Meir and V. Totik considered the ϕ -norm on C2π (the space 2π - periodic continuous functions) and estimated the deviation An (f)−f ϕ in terms of the modulus of continuity of f ∈ C2π , where (An) is a sequence of convolution operators from C2π into itself and ϕ is an increasing function on (0,∞) (Acta Math. Hung. 45 (1985), 441-443). In the present paper, an analogue of the theorem of Leindler, Meir and Totik is proved for functions periodic with respect to the hexagon lattice. Also, this theorem is applied to obtain estimates for approximation by partial sums of hexagonal Fourier series in H ¨older and generalized H ¨older norms.