Approximation of Bivariate Functions via Smooth Extensions

A Sharp Upper Bound on the Approximation Order of Smooth Bivariate Pp Functions

Computing with functions in two dimensions

New numerical methods are proposed for computing with smooth scalar and vector valued functions of two variables defined on rectangular domains. Functions are approximated to essentially machine precision by an iterative variant of Gaussian elimination that constructs near-optimal low rank approximations. Operations such as integration, differentiation, and function evaluation are particularly ...

Approximation of piecewise smooth functions and images by edge-adapted (ENO-EA) nonlinear multiresolution techniques

This paper introduces and analyzes new approximation procedures for bivariate functions. These procedures are based on an edge-adapted nonlinear reconstruction technique which is an intrinsically two-dimensional extension of the essentially non-oscillatory and subcell resolution techniques introduced in the one dimensional setting by Harten and Osher. Edge-adapted reconstructions are tailored t...

On Smooth Multivariate Spline Functions

In this paper the dimensions of bivariate spline spaces with simple cross-cut grid partitions are determined and expressions of their basis functions are given. Consequently, the closures of these spaces over all partitions of the same type can be determined. A somewhat more detailed study on bivariate splines with rectangular grid partitions is included. The results in this paper can be applie...

A sharp upper bound on the approximation order of smooth bivariate pp functions

A first result along these lines was given in [BH831] where it was shown that the approximation order of Π3,∆ (with ∆ the three-direction mesh) is only 3, which was surprising in view of the fact that all cubic polynomials are contained locally in this space. [J83] showed the corresponding result for C-quartics on the three-direction mesh and [BH832] provided upper and lower bounds for the appr...