A Convergence Result in Nodal Spline Interpolation

Peano Kernel Error Analysis for Quadratic Nodal Spline Interpolation

“…Also, from (4. which, together with the above mentioned significant improvement obtained for uniform partitions, clearly illustrates the usefulness of Peano kernel techniques for interpolation error analysis as opposed to the cruder estimation methods based on estimating the Lebesgue constant &V& , as employed in [7,9].…”

“…69 81], to obtain, for f # C r [a, b], r=1, 2, 3, error constants in the Jackson-type bounds for & f&Vf & which are, in particular for the uniform primary knot sequence (1.14), significantly smaller than the corresponding error constants obtained in [7,9].…”

“…In this paper we consider specifically the quadratic nodal spline interpolation error, and proceed to show how a Peano kernel technique can be used to establish Jackson-type estimates for the maximum norm error in which the error constants are, at least for the case of uniformly distributed spline knots, significantly smaller than those previously calculated by different methods in [7,9]. First, we introduce some notation, followed by a summary of those results from [5 9] which will be needed in our work.…”

“…In [7,9], de Villiers and Rohwer employed a method based on estimating the Lebesgue constant &V& to establish, for f # C r [a, b], r=0, 1, 2, 3, Jackson-type bounds on the interpolation error & f&Vf & , in which the explicit dependence on the local primary mesh ratio parameter R was explicitly calculated.…”

Martensen splines and finite-part integrals

“…In [4] two examples of sequences { f N } based on locally uniform partitions and satisfying (4)-(6) are provided for any positive integer p. These are the modified approximating splines and the modified optimal nodal splines, which are obtained by modifying the approximating splines [8] as well as the optimal nodal splines [1,2,3] in such a way that condition (5) is true for any positive integer p. In this paper, we consider sequences of approximating splines for which we can prove (4)-(6) without modifying their definition on [a, b]. In particular, we shall consider the Martensen spline operator, introduced in [9] and recently studied in [15,16].…”