On characterization of poised nodes for a space of bivariate functions

TL;DR

This paper investigates the characterization of poised nodes in bivariate function spaces, focusing on piecewise linear functions, and introduces a reduction method to determine poised sets.

Contribution

It provides the first characterization method for poised nodes in bivariate piecewise linear function spaces using a novel reduction approach.

Findings

A reduction method for identifying poised node sets in bivariate spaces

Characterization of poised nodes for bivariate piecewise linear functions

Extension of univariate interpolation results to bivariate case

Abstract

There are several examples of spaces of univariate functions for which we have a characterization of all sets of knots which are poised for the interpolation problem. For the standard spaces of univariate polynomials, or spline functions the mentioned results are well-known. In contrast with this there are no such results in the bivariate case. As an exception one may consider only the Pascal classic theorem, in the interpolation theory interpretation. In this paper we consider a space of bivariate piecewise linear functions, for which we can readily find out whether the given node set is poised or not. The main tool we use for this purpose is the reduction by a basic subproblem, introduced in this paper.