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

Abstract

Exact order estimates for some characteristics of linear and nonlinear approximation of the isotropic Nikol'skii-Besov classes $\mathbf{B}^r_{p,\theta}$ of periodic multivariate functions in the spaces $B_{q,1}$, $1\leq q \leq \infty$, are obtained. Among them are the best orthogonal trigonometric approximations, best $m$-term trigonometric approximations, Kolmogorov, linear and trigonometric widths. For all considered characteristics, their estimates coincide in order with the corresponding estimates in the spaces $L_{q}$. Moreover, the obtained exact in order estimates (except the case $1<p<2\leq q < \frac{p}{p-1}$) are realized by the approximation of functions from the classes ${\mathbf{B}}^r_{p,\theta}$ by trigonometric polynomials with the spectrum in cubic regions. In any case, they do not depend on the smoothness parameter $\theta$.

Keywords

• nikol'skii-besov class
• best orthogonal trigonometric approximation
• best approximation
• width