Gelfand Numbers of Embeddings of Mixed Besov Spaces

TL;DR

This paper investigates the asymptotic behavior of Gelfand numbers for embeddings of mixed Besov spaces into Lebesgue spaces, providing precise convergence rates and comparing with approximation numbers.

Contribution

It determines the correct order of Gelfand numbers for these embeddings in most cases and compares them with approximation numbers, advancing understanding of approximation complexity.

Findings

Established the asymptotic order of Gelfand numbers for the embeddings.

Compared Gelfand numbers with approximation numbers to highlight differences.

Provided results for almost all cases of the embedding parameters.

Abstract

Gelfand numbers represent a measure for the information complexity which is given by the number of information needed to approximate functions in a subset of a normed space with an error less than $\varepsilon$. More precisely, Gelfand numbers coincide up to the factor 2 with the minimal error $ e^{\rm wor}(n,\Lambda^{\rm all})$ which describes the error of the optimal (non-linear) algorithm that is based on $n$ arbitrary linear functionals. This explain the crucial role of Gelfand numbers in the study of approximation problems. Let $S^t_{p_1,p_1}B((0,1)^d)$ be the Besov spaces with dominating mixed smoothness on $(0,1)^d$. In this paper we consider the problem ${\rm App}: S^t_{p_1,p_1}B((0,1)^d) \to L_{p_2}((0,1)^d)$ and investigate the asymptotic behaviour of Gelfand numbers of this embedding. We shall give the correct order of convergence of Gelfand numbers in almost all cases. In addition we shall compare these results with the known behaviour of approximation numbers which coincide with $ e^{\rm wor-lin}(n,\Lambda^{\rm all})$ when we only allow linear algorithms.