ABSTRACT
We consider the problem of best approximation of a function by finite linear combinations of orthonormal wavelets with the error measured in a standard distribution space. We show that for a large class of spaces an algorithm based only on the size of the coefficients is essentially optimal. This allows us to sharpen some earlier results on nonlinear approximation by wavelets and to establish some new characterizations of functions with a given degree of approximation. We relate these results to the classical problem of comparing the Lebesgue space Lp to the sequence space lp , and obtain best possible imbeddings in terms of the discrete Lorentz spaces. We also consider compression of operators acting on certain Besov spaces, and characterize the operators that can be compressed well in terms of the size of the (rectangular) wavelet coefficients of the operator kernels.
