Multidimensional Chebyshev spaces, hierarchy of infinite-dimensional spaces and Kolmogorov-Gelfand widths
This work provides a foundational mathematical framework for multidimensional signal analysis and compressed sensing, potentially impacting theory of functions and signal processing.
The paper introduces Multidimensional Chebyshev spaces as a generalization of one-dimensional Chebyshev systems, defines a new hierarchy of infinite-dimensional spaces, and computes Kolmogorov-Gelfand widths for ellipsoidal function sets in multidimensional domains, solving a longstanding problem.
Recently the theory of widths of Kolmogorov (especially of Gelfand widths) 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, and their progress will have a major impact over analogous problems in the theory of multidimensional Signal Analysis. The present paper has three major contributions: 1. We solve the longstanding problem of finding multidimensional generalization of the Chebyshev systems: we introduce Multidimensional Chebyshev spaces, based on solutions of higher order elliptic equation, as a generalization of the one-dimensional Chebyshev systems, more precisely of the ECT--systems. 2. Based on that we introduce a new hierarchy of infinite-dimensional spaces for functions defined in multidimensional domains; we define corresponding generalization of Kolmogorov's widths. 3. We generalize the original results of Kolmogorov by computing the widths for special "ellipsoidal" sets of functions defined in multidimensional domains.