Dual basis functions in subspaces of inner product spaces

Dual basis functions are well-studied in the literature for certain inner product spaces. In this paper, we introduce dual basis functions in subspaces of inner product spaces. The goal is to construct a basis for a subspace that is dual to a basis in a different subspace of the same dimension. This problem reduces to the standard dual basis problem when the two subspaces and bases are the same.The paper begins with a characterization and properties of dual basis in subspaces, including requirements for existence. Then, the construction is carried out for subspaces of the space of polynomials in the Bernstein basis. Two configurations are of particular interest: a symmetric case in which the dual basis is affine and converges to Lagrange polynomial interpolation, and an end-point case that converges to Hermite interpolation.