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On the entropy numbers of the mixed smoothness function classes. (English) Zbl 1366.41019
The author studies the behavior of the entropy numbers of classes of multivariate functions with mixed smoothness. A new method of proving the upper bounds for the entropy numbers is developed, the method being based on the nonlinear approximation results, in particular, on greedy approximation. For the lower bounds the volume estimates method is used, which is a well-known method for proving the lower bounds for the entropy numbers. A detailed discussion of known results and open questions is also given.

MSC:
41A46 Approximation by arbitrary nonlinear expressions; widths and entropy
41A63 Multidimensional problems (should also be assigned at least one other classification number from Section 41-XX)
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