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Tree wavelet approximations with applications. (English) Zbl 1081.42031

In the first part of this interesting paper, the authors present a constructive greedy scheme (CGS) which generates a partition of an invariant set according to a given multivariate function. The partition is then used to construct a piecewise polynomial approximation to a given multivariate function. Optimal order of convergence is obtained for Sobolev and Besov norms. Tree wavelet approximation is an important development of nonlinear approximation. In the second part of this paper, the authors develop tree wavelet approximations by using the partition generated by CGS which have optimal order of convergence. A difficulty in the construction of such an approximation is to obtain a suitable index set. The authors show that this can be done by a tree index set associated with the partition generated by CGS. They provide sufficient conditions on a tree index set and on biorthogonal wavelet bases which ensure optimal order of convergence for wavelet approximations. Finally, some numerical examples are given.

MSC:

42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
46E35 Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems
65T60 Numerical methods for wavelets
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