The Breadth-one $D$-invariant Polynomial Subspace

TL;DR

This paper establishes the equivalence of two classes of breadth-one D-invariant polynomial subspaces and solves the discrete approximation problem in ideal interpolation for these subspaces, identifying specific points for coalescence.

Contribution

It demonstrates the equivalence of two representations of breadth-one D-invariant polynomial subspaces and solves the discrete approximation problem in ideal interpolation for these subspaces.

Findings

Proves the equivalence of two classes of D-invariant polynomial subspaces.

Provides a solution to the discrete approximation problem in ideal interpolation.

Identifies points where evaluation functionals converge to the induced functional space.

Abstract

We demonstrate the equivalence of two classes of $D$-invariant polynomial subspaces introduced in [8] and [9], i.e., these two classes of subspaces are different representations of the breadth-one $D$-invariant subspace. Moreover, we solve the discrete approximation problem in ideal interpolation for the breadth-one $D$-invariant subspace. Namely, we find the points, such that the limiting space of the evaluation functionals at these points is the functional space induced by the given $D$-invariant subspace, as the evaluation points all coalesce at one point.