Nonlinear wavelet approximation in anisotropic besov spaces

Abstract

Given anisotropic wavelet decompositions associated with the smoothness β, β = (β 1 ,...,β d ),β 1 ,...,β d > 0 of multivariate functions as measured in anisotropic Besov spaces B β , we give the rate of nonlinear approximation with respect to the L p -norm, 1 ≤ p < ∞, of functions f ∈ B β by these wavelets. We also prove that, among a general class of anisotropic wavelet decompositions of a function f E B β , the anisotropic wavelet decomposition associated with β gives the optimal rate of compression of the wavelet decomposition of f.