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

AbstractWe 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 σmnγδ(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 σmnγδ1,0 to 1,0 is worse in general than that by σmnγδ1,1 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