The Best Approximation For Some Classes Of Smooth Functions

Nonlinear approximation is one of the important parts of approximation theory.Generally speaking, the nonlinear approximation is superior to the linearapproximation. Its main advantage is that one can get the higher asymptotic orders tofunctions with less smoothness in the nonlinear case than in the corresponding linearapproximation. Nonlinear approximation is investigated extensively in applications,such as image and signal processing, noise removal, the numerical solution ofintegral equations and statistical estimation and so on.The best m-term approximation is a special form of nonlinear approximation. Itsbasic idea is that the approximants do not come from a fixed linear space but from anonlinear manifold, and the approximation space depends on the functions beingapproximated.This paper mainly studies the best m-term approximation for numerical normanisotropic Besov class and vectorial norm anisotropic Besov class with regard toorthogonal dictionaries. The orthogonal dictionary is commonly used because of itsgood properties. In order to obtain the upper estimate of the best m-termapproximation with regard to the orthogonal dictionary, we discuss approximate errorof these two kinds of classes with respect to the greedy algorithm at the same time.The full article is divided into four chapters. The first one is the introduction. Inthe second chapter and third chapter, we discuss the best m-term approximation fornumerical norm anisotropic Besov class and vectorial norm anisotropic Besov class,respectively. The fourth is the conclusion and prospect. First, we give therepresentation theorems by splitting the functions into blocks. Then we get the upperestimate for these two anisotropic classes with regard to the greedy algorithm and theasymptotic orders of the best m-term approximation applying with the basic relationsand analytical skills, such as representation theorems, Littlewood-Paley theorem andMarcinkiewicz type theorem. We consider this problem in the periodic multivariatecase for anisotropic Besov class of functions with vectorial norm. Due to thedifference of the smoothness and the measurement in every direction, there are manydifficulties to be solved. Therefore, the conclusions show that the asymptotic order ofthe best m-term approximation for numerical norm anisotropic Besov class withregard to orthogonal dictionaries can be achieved by a simply greedy algorithm.