Nonlinear Nonnested Spline Approximation

Abstract

Nonlinear approximation from regular piecewise polynomials (splines) supported on rings in \(\mathbb {R}^2\) is studied. By definition, a ring is a set in \(\mathbb {R}^2\) 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 that allow us to establish sharp inverse estimates in terms of Besov spaces.