Approximation of classes $B_{p,\theta }^r$ of periodic functions of one and several variables

Abstract

We obtain order-sharp estimates of best approximations to the classes \(B_{p,\theta }^r\) of periodic functions of several variables in the space Lq, 1 ≤ p, q ≤ ∞ by trigonometric polynomials with “numbers” of harmonics from step hyperbolic crosses. In the one-dimensional case, we establish the order of deviation of Fourier partial sums of functions from the classes \( B_{1,\theta }^{r_1 } \) in the space L1.