Wavelet decompositions for functions on a differentiable manifold
Wavelet decompositions for equidistant grid are well-known. In the case of irregular grid the wavelets have been studied before. The treatment of number currents connected with a differentiable manifold is possible with local functions, but creation of effective algorithms requires invoking the wavelet devices. The aim of the paper is to present the scheme of wavelet decompositions using the approximation relations. Here sufficient conditions of embedding of spaces for local functions on a differentiable manifold are discussed, the wavelet expansions and decomposition/reconstruction formulas are constructed, the estimates for the number of arithmetic operations and estimates of values of wavelet components are done. Classic approaches are connected with using the Fourier transform or with application of a lifting scheme. In the paper the approximation relations are initial; the relations mentioned lead to the wavelet expansions, which have asymptotically optimal order of approximation (with respect to N-width of standard compact sets). Besides supports of coordinate wavelets have minimal size (for prescribed order of approximation); in many cases the coordinate wavelets are coordinate splines, and their mean value (in contrast to classic ones) can be nonzero. Note also that the order of smallness of a wavelet component equals to the order of approximation, and the coefficient of linear dependence of calculating complexity on the input data is simply estimated with the help of the approximation order.