The Hermite interpolation
Let \(f \in C^ 1[- 1,1]\) with the usual norm \(\max (\| f \|_ \infty, \| f' \|_ \infty)\) and let \(H_{2n} (f)\) be the Hermite interpolation polynomial of degree at most \(2n - 1\) interpolating \(f\) and \(f'\) at the zeros \(x_ k\), \(k = 1, \dots, n\) of the Jacobi polynomial with weight \((1 - x)^ \alpha (1 + x)^ \beta\), \(\alpha, \beta > - 1\), \(x \in (-1,1)\). While for Lagrange interpolation at the nodes \(x_ k\) the norm of the Lagrange operator (Lebesgue constant) is asymptotically \(\log n\) for \(\alpha\), \(\beta \geq 1/2\), the norm of the operator \(H_{2n}\) is known not to be asymptotically \(\log n\) for every choice of the parameter \(\alpha\) and \(\beta\). In the present paper the authors use as nodes for Hermite interpolation the zeros of generalized Jacobi polynomials together when further points in \([-1, +1]\) (which increases the degree of the interpolating polynomial) and a weighted norm for \(C[-1,1]\) with generalized Jacobi weights. In this way they obtain completely analogous results to Lagrange interpolation. The paper is very clear written and contains besides this many interesting details also concerning simultaneous approximation.
scientific article; zbMATH DE number 739151
scientific article