On a hierarchy of infinite-dimensional spaces and related Kolmogorov-Gelfand widths

TL;DR

This paper introduces a new hierarchy of infinite-dimensional spaces based on higher order elliptic equations, extending Kolmogorov's original results and advancing the understanding of widths in multidimensional function and signal analysis.

Contribution

It generalizes Kolmogorov's widths to multidimensional spaces using elliptic equations, addressing fundamental gaps in the theory of widths for multidimensional functions.

Findings

Developed a hierarchy of infinite-dimensional spaces

Extended Kolmogorov's widths to multidimensional settings

Provided insights relevant to compressed sensing and signal sparsity

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 Compressed Sensing. It has been realized that widths reflect properly the sparsity of the data 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 by introducing a new hierarchy of infinite-dimensional spaces based on solutions of higher order elliptic equation.