International Journal of Mathematics and Mathematical Sciences
Volume 2008 (2008), Article ID 905635, 11 pages
doi:10.1155/2008/905635

Matrix transformations and disk of convergence in interpolation processes

Abstract

Let Aρ denote the set of functions analytic in |z|<ρ but not on |z|=ρ  (1<ρ<∞). Walsh proved that the difference of the Lagrange polynomial interpolant of f(z)∈Aρ and the partial sum of the Taylor polynomial of f converges to zero on a larger set than the domain of definition of f. In 1980, Cavaretta et al. have studied the extension of Lagrange interpolation, Hermite interpolation, and Hermite-Birkhoff interpolation processes in a similar manner. In this paper, we apply a certain matrix transformation on the sequences of operators given in the above-mentioned interpolation processes to prove the convergence in larger disks.