On optimal recovery in $L_2$

Abstract

We prove that the optimal error of recovery in the $L_2$ norm of functions from a class $\bF$ can be bounded above by the value of the Kolmogorov width of $\bF$ in the uniform norm. We demonstrate on a number of examples of $\bF$ from classes of functions with mixed smoothness that the obtained inequality provides a powerful tool for estimating errors of optimal recovery.

References in corpus (5)

• Integral norm discretization and related problems
• The Marcinkiewicz-type discretization theorems for the hyperbolic cross polynomials
• 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

Cited by in corpus (3)

• Bounds on Kolmogorov widths and sampling recovery for classes with small mixed smoothness
• Function values are enough for $L_2$-approximation: Part II
• Sampling discretization and related problems