On a C. de Boor's Conjecture in a Particular Case and Related Perturbation

TL;DR

This paper proves a specific case of C. de Boor's conjecture by showing that certain ideal projectors are limits of Lagrange projectors, using matrix computation and perturbation methods.

Contribution

It introduces a new class of D-invariant polynomial subspaces and demonstrates that ideal projectors associated with these are pointwise limits of Lagrange projectors, confirming a particular case of the conjecture.

Findings

Every ideal projector with the specified D-invariant subspaces is a limit of Lagrange projectors.

A concrete perturbation procedure for these ideal projectors is provided.

The result verifies a specific case of C. de Boor's conjecture.

Abstract

In this paper, we focus on two classes of D-invariant polynomial subspaces. The first is a classical type, while the second is a new class. With matrix computation, we prove that every ideal projector with each D-invariant subspace belonging to either the first class or the second is the pointwise limit of Lagrange projectors. This verifies a particular case of a C. de Boor's conjecture asserting that every complex ideal projector is the pointwise limit of Lagrange projectors. Specifically, we provide the concrete perturbation procedure for ideal projectors of this type.