Birkhoff Interpolation with Rectangular Sets of Nodes

TL;DR

This paper investigates Birkhoff interpolation on rectangular node sets, aiming to understand solvability through geometric criteria and conjectures, and highlights the need for advanced mathematical tools for a complete theory.

Contribution

It provides regularity criteria for Birkhoff interpolation on rectangular grids and discusses conjectures that extend understanding beyond existing methods.

Findings

Identified shapes with unique Lagrange problem solutions

Developed regularity criteria for solvability

Proposed conjectures for higher-dimensional cases

Abstract

Although it is important both in theory as well as in applications, a theory of Birkhoff interpolation with main emphasis on the shape of the set of nodes is still missing. Although we will consider various shapes (e.g. we find all the shapes for which the associated Lagrange problem has unique solution), we concentrate on one of the simplest shapes:``rectangular'' (also called "cartesian grids"). The ultimate goal is to obtain a geometrical understanding of the solvability. We partially achieve this by describing several regularity criteria, which we illustrate by many examples. At the end we discuss several conjectures which, we think, are important in understanding the behaviour of Birkhoff interpolation schemes in higer dimensions. Although we prove these conjectures in many unrelated cases, we believe that a ``complete proof'' requires new ideas which go beyond the usual methods in interpolation theory (and may reach areas such as algebraic geometry or algebraic topology).