The least solution for the polynomial interpolation problem (original) (raw)

We consider the following problem: given a subspace Λ of the dual Π of the space Π of svariate polynomials, find a space P ⊂ Π which is correct for Λ in the sense that each continuous linear functional on Λ can be interpolated by a unique p ∈ P . We provide a map, Λ → Λ ↓ ⊂ Π, which we call the least map, that solves this interpolation problem and give a comprehensive discussion of its properties. This least solution, Λ ↓ , is a homogeneous space and is shown to have minimal degree among all possible solutions. It is the unique minimal degree solution which is dual (in a natural sense) to all minimal degree solutions. It also interacts nicely with various maps applied to Λ, such as convolution, translation, change of variables, and, particularly, differentiation. Our approach is illustrated by detailed examples, concerning finite-dimensional Λ's spanned by point-evaluations or line integrals. Methods which facilitate the identification of the least solution are established. The paper is complemented by [BR3], in which an algorithmic approach for obtaining Λ ↓ is presented whose computational aspects are detailed.