Nonlinear nonnested 2-D spline approximation

TL;DR

This paper investigates nonlinear 2-D spline approximation on rings without nested structures, establishing Bernstein-type inequalities and inverse estimates in Besov spaces for smooth, nonnested splines.

Contribution

It introduces a new framework for nonlinear spline approximation on rings in ^2, proving Bernstein inequalities and inverse estimates without assuming nested ring structures.

Findings

Established Bernstein type inequalities for nonnested 2-D splines.

Proved sharp inverse estimates in Besov spaces for these splines.

Demonstrated approximation properties for smooth splines on rings.

Abstract

Nonlinear approximation from regular piecewise polynomials (splines) of degree $<k$ supported on rings in $\R^2$ is studied. By definition a ring is a set in $\R^2$ obtained by subtracting a compact convex set with polygonal boundary from another such a set, but without creating uncontrollably narrow elongated subregions. Nested structure of the rings is not assumed, however, uniform boundedness of the eccentricities of the underlying convex sets is required. It is also assumed that the splines have maximum smoothness. Bernstein type inequalities for this sort of splines are proved which allow to establish sharp inverse estimates in terms of Besov spaces.