ASYMPTOTICS OF APPROXIMATION OF FUNCTIONS BY CONJUGATE POISSON INTEGRALS

Among the actual problems of the theory of approximation of functions one should highlight a wide range of extremal problems, in particular, studying the approximation of functional classes by various linear methods of summation of the Fourier series. In this paper, we consider the well-known Lipschitz class Lip(1)alpha, i.e. the class of continuous 2 pi-periodic functions satisfying the Lips-chitz condition of order alpha, 0 < alpha <= 1, and the conjugate Poisson integral acts as the approximating operator. One of the relevant tasks at present is the possibility of finding constants for asymp-totic terms of the indicated degree of smallness (the so-called Kolmogorov-Nikol'skii constants) in asymptotic distributions of approximations by the conjugate Poisson integrals of functions from the Lipschitz class in the uniform metric. In this paper, complete asymptotic expansions are obtained for the exact upper bounds of deviations of the conjugate Poisson integrals from functions from the class Lip(1)alpha. These expansions make it possible to write down the Kolmogorov-Nikol'skii constants of the arbitrary order of smallness.