Dual Basis Functions in Subspaces

TL;DR

This paper investigates dual basis functions in polynomial subspaces, focusing on their construction, properties, and approximation capabilities, with potential applications to spline and multivariate polynomial spaces.

Contribution

It introduces a new approach to constructing dual bases in polynomial subspaces, emphasizing affine bases and their approximation properties.

Findings

Constructed dual bases that are affine and symmetric.

Derived approximation results involving quasi-interpolation.

Established convergence to Lagrange polynomial bases.

Abstract

In this paper we study dual bases functions in subspaces. These are bases which are dual to functionals on larger linear space. Our goal is construct and derive properties of certain bases obtained from the construction, with primary focus on polynomial spaces in B-form. When they exist, our bases are always affine (not convex), and we define a symmetric configuration that converges to Lagrange polynomial bases. Because of affineness of our bases, we are able to derive certain approximation theoretic results involving quasi-interpolation and a Bernstein-type operator. In a broad sense, it is the aim of this paper to present a new way to view approximation problems in subspaces. In subsequent work, we will apply our results to dual bases in subspaces of spline and multivariate polynomial spaces, and apply this to the construction of blended function approximants used for approximation in the sum of certain tensor product spaces.