Approximation of the Classes of Periodic Functions of One and Many Variables from the Nikol'skii–Besov and Sobolev Spaces.

We establish the exact-order estimates for the best orthogonal trigonometric approximations of the Nikol'skii–Besov classes B 1 , θ r (T d), 1 ≤ θ ≤ ∞, of periodic functions of one and many variables with predominant mixed derivative in the space Bθ, 1 (T d). In the multidimensional case, d ≥ 2, we establish the exact-order estimates for the approximations of the indicated classes of functions by their step-hyperbolic Fourier sums and determine the orders of orthoprojection widths in the same space. The behaviors of the corresponding approximation characteristics of the Sobolev classes W 1 , α r (T d) with d ∈ {1, 2} are also investigated. [ABSTRACT FROM AUTHOR]