Recently the theory of widths of Kolmogorov (especially of Gelfand widths)has received a great deal of interest due to its close relationship with thenewly born area of Compressed Sensing. It has been realized that widths reflectproperly the sparsity of the data in Signal Processing. However fundamentalproblems of the theory of widths in multidimensional Theory of Functions remainuntouched, and their progress will have a major impact over analogous problemsin the theory of multidimensional Signal Analysis. The present paper has threemajor contributions: 1. We solve the longstanding problem of findingmultidimensional generalization of the Chebyshev systems: we introduceMultidimensional Chebyshev spaces, based on solutions of higher order ellipticequation, as a generalization of the one-dimensional Chebyshev systems, moreprecisely of the ECT--systems. 2. Based on that we introduce a new hierarchy ofinfinite-dimensional spaces for functions defined in multidimensional domains;we define corresponding generalization of Kolmogorov's widths. 3. We generalizethe original results of Kolmogorov by computing the widths for special"ellipsoidal" sets of functions defined in multidimensional domains.
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