Optimal local spline interpolants

Sciarra

2014

Numer Algor

We state a uniform convergence theorem for finite-part integrals which are derivatives of weighted Cauchy principal value integrals. We prove that a sequence of Martensen splines, based on locally uniform meshes, satisfies the sufficient conditions required by the theorem. We construct the quadrature rules based on such splines and illustrate their behaviour by presenting some numerical results and comparisons with composite midpoint, Simpson and Newton-Cotes rules.

Section: G(x + H) − G(x)| G ∈ C(j)mentioning

confidence: 99%

Martensen splines and finite-part integrals

Sciarra

2014

Numer Algor

“…Employing a procedure based on the introduction of additional knots, De Villiers and Rohwer [4,5] constructed, for arbitrary order, an optimal nodal spline approximation operator W which was indeed shown to possess these three desired properties. Similar approaches have been followed for quadratic splines in [8] and for arbitrary order splines in [1].…”

Section: Introductionmentioning

confidence: 96%

“…However, it was shown in [4] that, in the case where the knots of the spline space are chosen to coincide with the interpolation points, the two properties of locality and interpolation are incompatible for quadratic or higher order splines.…”

Section: Introductionmentioning

confidence: 98%

Convergence of Derivatives of Optimal Nodal Splines

Journal of Approximation Theory

1997

Optimal nodal spline interpolants Wf of order m which have local support can be used to interpolate a continuous function f at a set of mesh points. These splines belong to a spline space with simple knots at the mesh points as well as at m&2 arbitrary points between any two mesh points and they reproduce polynomials of order m. It has been shown that, for a sequence of locally uniform meshes, these splines converge uniformly for any f # C as the mesh norm tends to zero. In this paper, we derive a set of sufficient conditions on the sequence of meshes for the uniform convergence of D j

“…However, when the knots of the spline space are chosen to coincide with interpolation points, the properties of locality and interpolation are incompatible for quadratic or higher degree splines [2].…”

Section: Introductionmentioning

confidence: 99%

“…Using a procedure based on the introduction of additional knots, De Villiers and Rohwer [2] constructed, for arbitrary order, an optimal nodal spline interpolation operator possessing the three desired properties.…”

Section: Introductionmentioning

confidence: 99%

Smoothness and error bounds of Martensen splines

Journal of Computational and Applied Mathematics

Sciarra

2015

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.