Ideal Interpolation

A linear interpolation scheme is termed ‘ideal’ when its errors form a polynomial ideal. The paper surveys basic facts about ideal interpolation and raises some questions. Ideal interpolation is, by definition, given by a linear projector on the space Π of polynomials whose kernel is a polynomial ideal. It is therefore also any linear map, as used in algebra, that associates a polynomial with its normal form with respect to a polynomial ideal. This article lists (and mostly proves) basic facts about ideal interpolation and raises some questions. §1. Definition and Basic Algebraic Facts If P is a linear projector of finite rank on the linear space X over the commutative field IF with algebraic dual X , then we can think of it as providing a linear interpolation scheme on X : For each g ∈ X , f = Pg is the unique element of ranP := P (X) for which λf = λg, ∀λ ∈ ranP ′ = {λ ∈ X ′ : λP = λ}, with P ′ the dual of P , i.e., the linear map X ′ → X ′ : λ 7→ λP . In other words, given that kerP := {g ∈ X : Pg = 0} = ran(id − P ), ranP ′ = (kerP ) := {λ ∈ X ′ : kerP ⊂ kerλ}, the set of interpolation conditions matched by P . Not surprisingly, there are exactly as many independent conditions as there are degrees of freedom, i.e., dim ranP = dim ranP . Proceedings Title 59 XXX (eds.), pp. 59–91. Copyright oc 200x by Nashboro Press, Brentwood, TN. ISBN 0-9728482-x-x All rights of reproduction in any form reserved.