Sharp error bounds for interpolating splines in tension
The authors consider splines in tension of an arbitrary partition of simple knots for the interval \([0, 1]\) with different nonnegative tension parameters on each subinterval. Error bounds are proven for the approximation of a function and its first three derivatives by two different types of interpolatory splines in tension and its respective derivatives: firstly, a \(C^1\) Hermite interpolant determined by function and first derivative values in the knots, and secondly, a \(C^2\) interpolant determined by function values in the knots and two additional endpoint derivative conditions. For these error bounds, the authors derive explicit constants dependent on the tension parameters. In the limiting cases of linear and cubic splines, the constants turn out to be the well-known exact ones for function value estimates by such polynomial splines. For the derivatives, the estimates, though not sharp, are improvements over known results.
scientific article; zbMATH DE number 825291