STRONG UNIFORM APPROXIMATION BY DOUBLE FOURIER-SERIES

We study the rate of strong uniform approximation to continuous functions f(x, y), 2π-periodic in each variable, by the rectangular partial sums of their double Fourier series. As special cases, we deduce strong approximation rates to functions in the Lipschitz classes Lip(α, β) and Zygmund classes Z(α, β), where α, β ε{lunate} (0, 1]. We also obtain the rates of strong uniform approximation to the conjugate functions f{hook} ∼(1,0), f{hook} ∼(0,1), and f{hook} ∼(1,1) by the rectangular partial sums of the corresponding conjugate series. With two exceptions, all rates are shown to be the best possible. © 1990.