ORDER ESTIMATES FOR THE BEST APPROXIMATIONS AND APPROXIMATIONS BY FOURIER SUMS OF THE CLASSES OF (ψ,β)-DIFFERENTIAL FUNCTIONS

摘要

We establish exact-order estimates for the best uniform approximations by trigonometric polynomials on the classes C_(β,p) ~ψ, of 2π-periodic continuous functions f defined by the convolutions of functions that belong to the unit balls in the spaces L_p, 1 ≤ p < ∞, with generating fixed kernels ψ_β ∈ L_p', 1/p + 1/p' =1. whose Fourier coefficients decrease to zero approximately as power functions. Exactorder estimates are also established in the L_p -metric, 1 ≤ p < ∞, for the classes C_(β,1) ~ψ of 2π-periodic functions f equivalent in terms of the Lebesgue measure to the convolutions of kernels ψ_β ∈ L_p with functions from the unit ball in the space L_1. It is shown that, in the investigated cases, the orders of the best approximations are realized by Fourier sums.