Infinite-dimensional generalization of Kolmogorov widths

TL;DR

This paper extends Kolmogorov widths to infinite-dimensional, multidimensional function spaces, addressing fundamental gaps in the theory relevant to signal processing and multidimensional analysis.

Contribution

It provides a novel multidimensional generalization of Kolmogorov's original result on widths of ellipsoidal sets of functions.

Findings

Generalized Kolmogorov widths to multidimensional function spaces

Addresses gaps in multidimensional signal analysis theory

Lays groundwork for future research in multidimensional compressive sensing

Abstract

Recently the theory of widths of Kolmogorov-Gelfand has received a great deal of interest due to its close relationship with the newly born area of Compressive Sensing in Signal Processing. However fundamental problems of the theory of widths in multidimensional Theory of Functions remain untouched, as well as analogous problems in the theory of multidimensional Signal Analysis. In the present paper we provide a multidimensional generalization of the original result of Kolmogorov about the widths of an "ellipsoidal sets" consisting of functions defined on an interval.