Work on nonlinear approximation and its applications to numerical problems will be undertaken in this project. This type of approximation has become increasingly important since it may give much smaller errors than traditional approximation methods. Emphasis will be placed on nonlinear methods in the general setting of wavelet decompositions with particular attention paid to applications to rational and spline approximations. Underlying this work is the role played by nonlinear approximations in the theory of nonlinear partial differential equations and in problems of image and surface compression. In nonlinear approximation, one replaces a linear space of approximating functions by a nonlinear manifold. Splines with free knots, rational functions and certain types of adaptive approximation are among the most studied examples. The advantage of nonlinear approximation, recognized for years, is that it allows for better approximation of functions with singularities. The point of view taken by researchers in this area is not to determine whether or not a given function can be approximated by functions taken from some class. Rather, one begins with the class and asks for a characterization of all functions which may be approximated within a prescribed error. Recent work shows that it is possible to identify these sets with known, identifiable, families of functions (characterized, for example, by their average oscillation). Much of the best work has been confined to one-dimesional approximation. This project will consider a relatively new approach to multivariate approximation. The traditional approach is one of breaking up the domain into suitable smaller domains on which approximation can be established. A more promising approach, using the wavelet concept, considers the partitioning function into more manageable parts which lend themselves to good approximation.
