Hermite-Lagrange interpolation and Schur's expansion of sin \(\pi\) x
With the help of Taylor's formula in the theory of right-invertible operators, a new constructive approach to the Hermite-Lagrange 2-point interpolation polynomial with explicit remainder and to Schur's expansion of sin \(\pi\) x is presented. This improve earlier results of I. Schur, G. C. Rota, D. Kahaner, A. Odlyzko, S. Wrigge, and A. Fransen. Also, we prove: If \(f\in C^{\infty}(0,1)\) is completely convex, then \(f(x)=\sum^{\infty}_{n=0}\frac{1}{n!}(a_ n+b_ nx)(x(1-x))^ n,\) \(a_ n\geq 0\), \(a_ n+b_ n\geq 0\), uniformly in (0,1).
Hermite-Lagrange 2-point interpolation polynomial
scientific article; zbMATH DE number 4077925
scientific article