Peano Kernel Error Analysis for Quadratic Nodal Spline Interpolation

Abstract

Using a Peano kernel technique, Jackson-type estimates with respect to the maximum norm are derived for the quadratic nodal spline interpolation error. The explicitly calculated error constants are shown to grow linearly with respect to the local mesh ratio parameter, and are, at least for the important special case of a uniform spline knot sequence, significantly smaller than those previously calculated by different methods. Academic Press

Gregory type quadrature based on quadratic nodal spline interpolation

“…It was also shown in [3] that, in the case of uniform partitions Π n and n in the sense of (1.3) and (2.1), the estimates (3.3) can be shown to be sharper than those obtained by merely setting R = 1 in (3.5) -(3.10): indeed, it was found (see equation (5.4) in [3]) that, specifically for the case when the bottom line of (2.1) is satisfied, the estimates (3.3) hold with the positive numbers M r given by In Sect . 3.2 below we proceed to demonstrate how the error estimates (3.3) can, at least for values of R which are close to one, and therefore in particular including the value R = 1 yielding the non-optimal order Lacroix rule error constants M r as given by (3.12), be sharpened significantly by using a Peano kernel technique specifically suited for quadrature error analysis.…”

“…In [3], a Peano kernel technique was used to obtain Jackson-type estimates for the quadratic nodal spline interpolation error ||f − V n f || ∞ . These estimates were an improvement on those previously established in [8] by means of a bound on the Lebesgue constant ||V n || ∞ , and are given in terms of the following mesh parameters.…”

“…The non-optimal order error bounds in (1.10) for f ∈ C r [a, b], r = 1, 2, 3, can be sharpened by employing improved estimates for the quadratic nodal spline interpolation error f − f ∞ , as recently obtained by means of a Peano kernel technique by de Swardt and de Villiers in [3], together with (1.8), to yield the bounds…”

Histopolating splines

“…Furthermore, since the key for high approximation power of the operator I n lies in an appropriate reproduction of polynomials, the operators having this property have been preferred in the literature. de Villiers and Rohwer, for example, studied nodal splines which combine the three desired features of interpolation, locality and high polynomial reproduction in [12][13][14], and de Villiers investigated their properties in [10,11], while de Swardt and de Villiers analyzed nodal splines in [8,9]. Nodal splines have also been studied by the author in [17].…”

“…1, the fundamental nodal cubic (n = 3) spline with respect to t = Z is depicted (above, left). For further references concerning nodal splines, see, for instance, [3,[8][9][10][11]13,14]. Note that in case of t j = j , the nodal spline L n,0 is even if and only if n is even.…”