Characteristics of linear and nonlinear approximation of isotropic classes of periodic multivariate functions

Keywords:
Nikol'skii-Besov class, best orthogonal trigonometric approximation, best approximation, widthAbstract
Exact order estimates for some characteristics of linear and nonlinear approximation of the isotropic Nikol'skii-Besov classes Brp,θBrp,θ of periodic multivariate functions in the spaces Bq,1Bq,1, 1≤q≤∞1≤q≤∞, are obtained. Among them are the best orthogonal trigonometric approximations, best mm-term trigonometric approximations, Kolmogorov, linear and trigonometric widths.
For all considered characteristics, their estimates coincide in order with the corresponding estimates in the spaces LqLq. Moreover, the obtained exact in order estimates (except the case 1<p<2≤q<pp−11<p<2≤q<pp−1) are realized by the approximation of functions from the classes Brp,θBrp,θ by trigonometric polynomials with the spectrum in cubic regions. In any case, they do not depend on the smoothness parameter θθ.