## 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.

### MSC:

 41A05 Interpolation in approximation theory 41A15 Spline approximation

uniqueness
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### References:

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