Near Best Tree Approximation

Tree approximation is a form of nonlinear wavelet approximation that appears naturally in applications such as image compression and entropy encoding. The distinction between tree approximation and the more familiar n-term wavelet approximation is that the wavelets appearing in the approximant are required to align themselves in a certain connected tree sturcture. This makes their positions easy to encode. Previous work [CDGO], [CDDD] has established upper bounds for the error of tree approximation for certain (Besov) classes of functions. The present paper, in contrast, studies tree approximation of individual functions with the aim of characterizing those functions with a prescribed approximation error. This is accomplished in the case that the approximation error is measured in L2, or in the case p 6= 2, in the Besov spaces B0 p(Lp), which is close to (but not the same as) Lp. Our characterization of functions with a prescribed approximation order in these cases is given in terms of a certain maximal function applied to the wavelet coefficients. AMS subject classification: tree approximation, nonlinear approximation, wavelets, approximation classes, adaptive approximation 41A25, 41A46, 65F99, 65N12, 65N55.