A Short Note on the Comparison of Interpolation Widths, Entropy Numbers, and Kolmogorov Widths

TL;DR

This paper compares various widths of compact operators, explores their relationships in different spaces, and identifies conditions under which specific gaps between these widths occur, especially in Sobolev spaces.

Contribution

It provides new insights into the relationships between interpolation, entropy, and Kolmogorov widths, including conditions for significant gaps in these measures.

Findings

A sufficient condition for a gap of order n^{1/2} between interpolation and Kolmogorov widths.

Identification of a gap between Kolmogorov and approximation widths in multi-dimensional Sobolev spaces.

Application of general results to embeddings between reproducing kernel Hilbert spaces and L_infinity spaces.

Abstract

We compare the Kolmogorov and entropy numbers of compact operators mapping from a Hilbert space into a Banach space. We then apply these general findings to embeddings between reproducing kernel Hilbert spaces and $L_\infty(\mu)$. Here we provide a sufficient condition for a gap of the order $n^{1/2}$ between the associated interpolation and Kolmogorov $n$-widths. Finally, we show that in the multi-dimensional Sobolev case, this gap actually occurs between the Kolmogorov and approximation widths.