Q-Splines

Abstract

The classical weighted spline introduced by Ph. Cinquin (1981), (see also K. Salkauskas (1984) and T.A. Foley (1986)) consists in minimizing ∫ ^b _a w(t)(x″(t))² dt under the conditionsx(t _i )=y _i ,i=1,...,n, where the functionw is piecewise constant on the subdivisiona<t ₁<t ₂<...<t _n <b. The solution is a cubic spline, but it is notC ². We consider here the minimization of

$$\int_a^b {\frac{{\left( {x''\left( t \right)} \right)^2 }}{{q\left( t \right)}}} dt$$

whereq is a piecewise polynomial function. We shall study in detail the case whenq is continuous and piecewise linear on the subdivision. The valuesq _i =q(t _i ) act as shape parameters. The solution is aC ² quartic spline, but surprisingly it has, in fact, all the advantages of the cubic spline, namely:

• - computing the solution leads to a symmetric tri-diagonal linear system,

• - computing the corresponding smoothing spline leads to a block 2×2 tri-diagonal system,

• - the associatedB-spline is based on 4 intervals of the subdivision (as in the case of the classical cubicB-spline).

The properties of this new weighted spline will be developed and its efficiency for interpolating, smoothing or designing will be illustrated on a selection of examples. Finally, a weighted spline withG ² continuity is described, which has three shape parameters at each knot.