Approximation to continuous functions by Cesàro means of double Fourier series and conjugate series

Abstract

We study the rate of uniform approximation to continuous functions ƒ(x, y), 2π-periodic in each variable, in Lipschitz classes Lip(α, β) and in Zygmund classes Z( α, β), 0 < α, β ⩽ 1, by Cesàro means σ mn γδ(ƒ) of positive orders of the rectangular partial sums of double Fourier series. The rate of uniform approximation to the conjugate functions ▪ 1,0, ▪ 0,1 and ▪ 1,1 by the corresponding Cesàro means is also discussed in detail. The difference between the classes Lip(α, β) and Z( α, β), similar to the one-dimensional case, appears again when max(α, β) = 1. (Compare Theorems 2 and 3 with Theorems 4 and 5.) One surprising result is the following: The uniform approximation rate by σ mn γδ ▪ 1,0 to ▪ 1,0 is worse in general than that by σ mn γδ ▪ 1,1 to ▪ 1,1 for ƒ ϵ Lip(1, 1) . In fact, the appearance of an extra factor [log( n + 2)] 2 in the former case is unavoidable (see Theorem 6). All approximation rates we obtain, with one exception, are shown to be exact. Two conjectures are also included.