On characterization of poised nodes for a space of bivariate functions
This work provides a practical tool for identifying poised node sets in a specific bivariate function space, but it is incremental as it only addresses a particular case and does not generalize to broader bivariate spaces.
The paper addresses the lack of characterization of poised node sets for bivariate interpolation, focusing on a space of bivariate piecewise linear functions. It introduces a reduction method using a basic subproblem to determine whether a given node set is poised.
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.