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}_{[a, b} (n⩾2, κ⩾0) in C[a, b]. Namely, it is shown that the following theorem is valid: for arbitrary numbers η₀, η₁, ..., η_n+k, satisfying the conditions (η_i−η_i−1) (η_i+1{ η _i< 0(i=1, ..., n +k−1), there is a unique polynomials _n,k (t)∈ ℊ ^]/n,k_{[a, b} and pointsa=ξ₀,<ξ₁<...<ξ _n+k− 1<ξ _n+k = b (ξ₁<x₁ <ξ_n, ..., ξ_k<x_k<ξ_n+k−1), such that s_n,k(ξ_i) = η_i(i=0, ..., n + k), s′_n,k(ξ_i)=0 (i=1, ..., n + k−1).