A Voronovskaya Theorem for Variation-Diminishing Spline Approximation

Abstract

In [7] Schoenberg introduced the following variation-diminishing spline approximation methods.Let m > 1 be an integer and let Δ = {xi} be a biinfinite sequence of real numbers with xi ≧ xi + l < xi+m. To a function f associate the spline function Vf of order m with knots Δ defined by(1.1)whereand the Nj(x) are B-splines with support xj < x < xj+m normalized so that ΣjNj(x) = 1. See, e.g., [2] for a precise definition of the Nj(x) and a discussion of the properties of Vf.