Spline polynomials with a prescribed sequence of extrema

Abstract

In the present note a theorem about strong suitability of the space of algebraic polynomials of degree ≤n in C[a,b] (Theorem A in [1]) is generalized to the space of spline polynomials ℊ [ ]n, k (n⩾2, κ⩾0) in C[a, b]. Namely, it is shown that the following theorem is valid: for arbitrary numbers η0, η1, ..., ηn+k, satisfying the conditions (ηi−ηi−1) (ηi+1{ η i< 0(i=1, ..., n +k−1), there is a unique polynomials n,k (t)∈ ℊ [ ]/n, and pointsa=ξ0,<ξ1<...<ξ n+k− 1<ξ n+k = b (ξ1<x1 <ξn, ..., ξk<xk<ξn+k−1), such that sn,k(ξi) = ηi(i=0, ..., n + k), s′n,k(ξi)=0 (i=1, ..., n + k−1).