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.
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.
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