On the asymptotic behavior of the optimal error of spline interpolation of multivariate functions

Abstract

The question of adaptive approximation by splines has been studied for a number of years by various authors. The results obtained have numerous applications in computational and discrete geometry, computer aided geometric design, finite element methods for numerical solutions of partial differential equations, image processing, and mesh generation for computer graphics, among others.

In this dissertation we investigate some questions of adaptive approximation by various classes of splines (linear, multilinear, biquadratic) with free knots. In particular, we will study the asymptotic behavior of the optimal error of weighted approximation in different norms by interpolating splines from these classes. The proofs lead to the construction of asymptotically optimal simplicial or box partitions of the domain for interpolation by linear and multilinear splines, respectively. In addition, applications to quadrature formulas are considered.