Topics in Nonlinear Approximation

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The main goal of the proposed research program is further investigation of the connection, in various settings, between smoothness properties of (classes of) functions and their approximation orders. The main categories of the program are: (A) polynomial and spline approximation: new moduli of smoothness and their properties, characterization of approximation orders of various classes via smoothness measured in terms of the new moduli of smoothness, weighted polynomial approximation (Jackson, Whitney and Bernstein-type theorems), simultaneous polynomial approximation with weights, equivalence of moduli of smoothness of polynomials and splines, and applications; (B) constrained approximation and interpolation: widths of constrained function classes; characterization of constrained approximation spaces, exact approximation orders of constrained function classes, multivariate constrained approximation, approximation by q-monotone functions, relaxing constraints. Some of specific research topics are: I. New moduli of smoothness and applications. In a series of papers published in 2014-2015, together with co-authors, the proposer introduced new moduli of smoothness, applied them to obtain Jackson-type estimates for approximation of functions in Lp by means of algebraic polynomials, and proved matching inverse theorems as well, thus obtaining constructive characterization of various smoothness classes of functions via the degree of their approximation by polynomials. These moduli are precisely what's needed for direct results for constrained approximation in the Lp (quasi)norms. Their analogs have been successfully used to establish various equivalence relations for several classes of functions in L\infty, but analogous problems for constrained approximation in Lp are still open and require new techniques. With the introduction of the new modulus it is now known what form these estimates should take. II. Weighted constrained and unconstrained approximation. Another topic of the proposed research is weighted polynomial approximation and weighted approximation of various classes of functions having certain shape properties. In 2014-2016, the proposer obtained matching direct and inverse theorems in all Lp, p>0, (quasi)norms weighted by certain averages of doubling weights depending on the degree of approximating polynomials, and analogous results were obtained for doubling weights having zeros and singularities (whose presence makes the approximation process significantly more difficult). There are numerous problems in this area that still need to be investigated. Additionally, there are almost no results on weighted approximation of various classes of functions having certain shape properties, and research on all of these topics is proposed to be continued.