Topics in Nonlinear Approximation

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One of the central tasks in Approximation Theory is to determine a connection between approximation and smoothness properties of what's been approximated (functions, surfaces, etc.), and the main goal of the proposed research program is further investigation of the exact connection between smoothness of (classes of) functions (surfaces, bodies, etc.) and their approximation orders. In general, nonlinear approximation methods perform much better than linear ones, and the order of approximation  from nonlinear manifolds is much better than the order of approximation by the elements of linear spaces, which is the main reason for investigating nonlinear approximation techniques. The proposed research program falls into the following main categories: (A) Polynomial and Spline Approximation: characterization of approximation spaces, measures of smoothness, weighted and unweighted approximation, simultaneous approximation, and applications; (B) Constrained Approximation and Interpolation: characterization of constrained approximation spaces (with and without weights), approximation orders, Kolmogorov, linear and pseudo-dimensional widths of constrained function classes, multivariate constrained approximation, Whitney type interpolatory results for k-monotone functions and applications, approximation by k-monotone functions, relaxing constraints; (C) Adaptive Algorithms: data adapted isotropic and anisotropic triangulations, convergence, error estimates, "optimal" mesh generation. Each of these categories consists of several subcategories. For example, some of the specific problems that the proposer is planning to investigate are: (i) New moduli of smoothness and their properties, (ii) Characterization of approximation orders of various classes via smoothness measured in terms of the new moduli of smoothness, (iii) Measure of smoothness with weights and Jackson and Whitney type theorems for weighted polynomial approximation, (iv) Simultaneous polynomial approximation with weights, (v) Multivariate constrained approximation by (piecewise linear) polynomials, (vi) Shape preserving approximation in (weighted) Lp norms.