Quadratic splines interpolating derivatives. (English) Zbl 0758.41005

Let \(\Delta_ x=\{x_ i: i=o(1)n+1\}\) be an increasing set of knots and let \(S(2,\Delta_ x)\) be the set of spline functions \(s\) satisfying the conditions: \(s\in C^ 1[x_ 0,x_{n+1}]\), \(s\) is a quadratic polynomial on each interval \([x_ i,x_{i+1}]\), \(i=o(1)n\). The author proves that there exists a unique spline function \(s\in S(2,\Delta_ x)\) verifying the conditions \(s'(x_ i)=m_ i\), \(i=o(1)n+1\) (interpolation conditions) and \(s(x_ 0)=s_ 0\) (the initial condition). One studies the existence and the uniqueness of a spline function \(s\in S(2,\Delta_ x)\) taking given values for the first and the second derivatives at the interpolation points \(t_ i\), \(i=o(1)n\), i.e. on the mesh \[ (\Delta_ x\Delta_ t): x_ 0\leq t_ 0<x_ 1<t_ 1<\cdots< t_ n\leq x_{n+1}, \] are verified conditions of the form \(s'(t_ i)=m_ i\) and \(s''(t_ i)=M_ i\), \(i=o(1)n\). The author gives algorithms for determining the parameters involved in the calculations of these splines.


41A05 Interpolation in approximation theory
41A15 Spline approximation


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