Spline Interpolation

Abstract

Spline interpolation is an improvement over piecewise — polynomial interpolation. It uses less information of the given function, yet furnishes smoother interpolates. The plan of this chapter is as follows: In Section6.2 we shall define the spline space S _m,τ (Δ), and for a given function x (t) the spline and Lidstone — spline interpolates S ^Δ_m,τ and LS ^Δ_m,2m−2 x(t),respectively. Here, we shall also show that to acquire the bounds for ‖ D ^k(x− S ^Δ_m,τ ) ‖∞ and ‖ D ^k(x− LS ^Δ_m,2m−2 x) ‖∞ in terms of the derivatives of x(t) it is necessary to estimate several terms. While some of these terms can be estimated by the results of Chapter 5, other terms which require a different analysis for each m and τ, need to be bounded. In Sections6.3,6.4 and 6.6 respectively, we shall consider the cases m = 2, τ = 2; m = 3, τ = 4 and m = 3, τ = 3. These cases correspond to cubic in the class C ⁽²⁾ [a, b], and quintic in the classes C ⁽⁴⁾[a, b] and C ⁽³⁾[a, b] spline interpolates. For the cases m = 3, τ = 4 and m = 3, τ = 3 in Sections 6.5 and 6.7 respectively, we shall discuss the construction of approximated quintic splines, and for these interpolates we will provide explicit error bounds in L _∞ — norm. For the cubic and quintic Lidstone — spline interpolates error bounds in L _∞ — norm are obtained in Sections 6.8 and 6.9, respectively. In Section6.10 we shall extend Theorems 5.3.16 and 5.3.17 for the spline interpolate S ^Δ_m,τ x(t)