Hermite-lagrange interpolation and Schur's expansion of sin πx

Abstract

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 πx is presented. This improves earlier results of I. Schur, L. Carlitz, G. C. Rota, D. Kahaner, A. Odlyzko, S. Wrigge, and A. Fransén. Also, we prove: If ƒ ϵ C ∞ [0, 1] is completely convex, then ƒ(x) = ∑ n = 0 ∞ 1 n! (a n + b nx)(x(1 − x)) n, a n ⩾0, a n+b n⩾0 uniformly in [0, 1 ].