Dual Bases Functions in Subspaces

In this paper we study dual bases functions in subspaces. These are bases which are dual to subsets of functionals from larger linear spaces. 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 and linear polynomial reproduction, we are able to derive certain approximation theoretic results involving quasi-interpolation and a Bernstein-type operator. We also apply our construction to splines and multivariate polynomials. In particular, we give characterizations in the multivariate setting in B-form for existence of dual bases.