Sharp Bounds for the Lebesgue Constant in Quadratic Nodal Spline Interpolation

Applications of optimally local interpolation to interpolatory approximants and compactly supported wavelets

“…(b) In the paper [15] it is shown that, for the optimal order (ρ = n) polynomial reproduction case, and with the definition (1.2) of x changed to x i := t (n−1)i , i ∈ Z, the minimally supported spline approximant, constructed similarly to those in §2.1 above, automatically interpolates at x, a result which is of course consistent with the quadratic case n = 3 of Theorem 2.3 above.…”

“…However, since we are interested in smallest possible and specific supports as designated by the index ν, great care has to be taken in the construction of the quasi-interpolants that give a tight match with the optimally local interpolation operator L n,ν . The techniques we will introduce are based on those in [14,15], where x i = t (n−1)i , n ≥ 3, in (1.2) was considered instead. (For further development of this approach, the reader is referred to [16,17].)…”

Convergence of Derivatives of Optimal Nodal Splines

“…In this section we give the necessary background material on optimal nodal spline interpolants based on the work in [6].…”

“…The approximation properties of W were studied in both [3] and [6]. In the present paper we will continue the investigation of De Villiers and Rohwer on how well the nodal spline Wf approximates a smooth function.…”

“…In Section 2 we shall introduce the nodal splines and report the convergence article no. AT963012 results from [3] and [6]. In Section 3 we give the conditions under which D j Wf converges uniformly to D j f, for f # C s and j=1, ..., s<m, where m is the order of the spline, and D j is the jth derivative operator.…”