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Best \(m\)-term approximation and Sobolev-Besov spaces of dominating mixed smoothness – the case of compact embeddings. (English) Zbl 1251.41005
In this paper, \(m\)-widths of tensor products of approximants separately from Sobolev and from Besov spaces are studied and asymptotically estimated, where the spaces are compactly embedded. The errors whose infima are taken are measured in the \(L_p\)-norm and the system from which the approximations are taken are tensor product wavelet systems with compactly supported wavelets and additional conditions on smoothness, moments and integrability. The principal general classes of function spaces which are used are the Besov-Lizorkin-Triebel spaces as those spaces which contain the mentioned Sobolev and Besov spaces (Chapter 3, and Chapter 5 for the associated sequence spaces) and the main theorems are summarised in Chapter 2. Among other things, the \(m\)-width results are also compared with best linear approximations and entropy numbers.

MSC:
41A25 Rate of convergence, degree of approximation
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)
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
46B70 Interpolation between normed linear spaces
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