Smoothness and error bounds of Martensen splines

Abstract

Martensen splines M f of degree n interpolate f and its derivatives up to the order n − 1 at a subset of the knots of the spline space, have local support and exactly reproduce both polynomials and splines of degree ≤ n. An approximation error estimate has been provided for f ∈ C n+1 . This paper aims to clarify how well the Martensen splines M f approximate smooth functions on compact intervals. Assuming that f ∈ C n−1 , approximation error estimates are provided for D j f, j = 0, 1, . . . , n − 1, where D j is the jth derivative operator. Moreover, a set of sufficient conditions on the sequence of meshes are derived for the uniform convergence of D j M f to D j f , for j = 0, 1, . . . , n − 1.

Martensen splines and finite-part integrals

“…Lemma 1 gives a local estimate for |D n M R ( f )(t)|, with t ∈ [t l , t l+1 ]. Lemma 2, proved in [5], provides a local estimate for |e (s) R (t)|, with t ∈ [t l , t l+1 ] and s = 0, 1, . .…”

“…In this section, we give the necessary background material on Martensen splines as presented in [15,16,17] and [5].…”

“…The following uniform convergence result is provided in [5] for Martensen splines and its derivatives.…”

“…We consider two different singularities of order 3, λ = t [R/4] +h/2+τh and λ = t 0 +h/2+τh, with τ = 0, 1/4, where h is the width of every subinterval. The exact value of the finite-part integral is [21] I( f ; λ; 2) = 6λ − 8λ 3 − 6λ 5 (1 − λ 2 ) 2 + 6λ 2 ln 1 − λ 1 + λ . In order to construct the composite Simpson's rule, we have to introduce a quadrature node at each subinterval then, to compare our method with Simpson, we have to assume R = 2n, where n is the number of subintervals used for the construction of the composite rule.…”