Bounds on Kolmogorov widths and sampling recovery for classes with small mixed smoothness

Abstract

Results on asymptotic characteristics of classes of functions with mixed smoothness are obtained in the paper. Our main interest is in estimating the Kolmogorov widths of classes with small mixed smoothness. We prove the corresponding bounds for the unit balls of the trigonometric polynomials with frequencies from a hyperbolic cross. We demonstrate how our results on the Kolmogorov widths imply new upper bounds for the optimal sampling recovery in the $L_2$ norm of functions with small mixed smoothness.

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• Integral norm discretization and related problems
• Constructive sparse trigonometric approximation and other problems for functions with mixed smoothness
• Entropy numbers and Marcinkiewicz-type discretization theorem
• Marcinkiewicz-type discretization of $L^p$-norms under the Nikolskii-type inequality assumption
• Sampling discretization of integral norms
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Cited by in corpus (3)

• A sharp upper bound for sampling numbers in $L_2$
• A note on sampling recovery of multivariate functions in the uniform norm
• Sampling discretization and related problems