On a hierarchy of infinite-dimensional spaces and related Kolmogorov-Gelfand widths
For researchers in approximation theory and compressed sensing, this provides a multidimensional extension of a foundational result, but the novelty is incremental as it builds on existing theory.
The paper generalizes Kolmogorov's width theory to multidimensional settings by introducing a hierarchy of infinite-dimensional spaces based on higher-order elliptic equations, addressing fundamental problems in function theory and signal analysis.
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.