Further asymptotic properties of best approximation by splines

Abstract

Let S n k, r denote the collection of polynomial splines of order k with at most ( n − 1) free knots, each of multiplicity r. This paper explicitly finds the constants C k, r, p so that lim n→∞ n k dist L p[0, 1] {ƒ, S n k,r} = C k,r,p ∥ƒ (k)∥ L σ[0,1] , where σ = p (kp + 1) and ƒ is sufficiently smooth. This completely fills the gap between previously known results for simple knots ( r = 1) and for piecewise polynomials ( r = k). We also consider similar asymptotic properties pertaining to approximation of vector functions by vector splines. We define the latter to be families of vector functions whose components are splines of order k with common knots.