Dual Basis Functions in Subspaces

In this paper, we present “dual bases functions in subspaces”. Suppose is a basis for an n-dimensional space X that is dual to some linear functionals and Y is a subspace of X. We are interested in bases for Y that are dual to “subsets” of assuming these subsets are linearly independent on Y. Our goal in this paper is to construct a general framework for computing dual bases in subspaces. Specifically, our interest is in bases that are “affine”, in the sense that they sum to 1, with primary focus on the construction of Bernstein-like bases for polynomial spaces. While our bases are affine, they are not convex (they are not positive on We show that in a certain symmetric configuration, where the subsets of are spaced out uniformly, the corresponding dual bases converge to the Lagrange polynomial basis as In the last part of paper, we apply our new basis to the problem of degree-reduction