  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.
