$L_p$-Sampling recovery for non-compact subclasses of $L_\infty$

TL;DR

This paper investigates the problem of sampling recovery for certain non-compact multivariate function classes in $L_$, introducing new bounds and methods for functions with very small smoothness in Besov and Triebel-Lizorkin spaces.

Contribution

It establishes uniform boundedness of Faber-Schauder coefficients and controls the approximation error in $L_q$, providing sharp convergence rates for non-compact subclasses.

Findings

Proves uniform boundedness of Faber-Schauder coefficients.

Controls the error of truncated Faber-Schauder series in $L_q$.

Main convergence rate is shown to be sharp.

Abstract

In this paper we study the sampling recovery problem for certain relevant multivariate function classes which are not compactly embedded into $L_\infty$. Recent tools relating the sampling numbers to the Kolmogorov widths in the uniform norm are therefore not applicable. In a sense, we continue the research on the small smoothness problem by considering "very" small smoothness in the context of Besov and Triebel-Lizorkin spaces with dominating mixed regularity. There is not much known on the recovery of such functions except of an old result by Oswald in the univariate situation. As a first step we prove the uniform boundedness of the $\ell_p$-norm of the Faber-Schauder coefficients in a fixed level. Using this we are able to control the error made by a (Smolyak) truncated Faber-Schauder series in $L_q$ with $q<\infty$. It turns out that the main rate of convergence is sharp. As a consequence we obtain results also for $S^1_{1,\infty}F([0,1]^d)$, a space which is ``close'' to the space $S^1_1W([0,1]^d)$ which is important in numerical analysis, especially numerical integration, but has rather bad Fourier analytic properties.