Hermite-Birkhoff Interpolation on Arbitrarily Distributed Data in Banach Spaces

Abstract

A class of cardinal basis functions is proposed in order to achieve a generalization to Banach spaces of Hermite-Birkhoff interpolation on arbitrarily distributed data. First, a constructive characterization of the class of cardinal basis functions is given. Then, the interpolation problem is solved by using a suitable combination of such functions and Taylor-Fréchet expansions. The performance of the obtained interpolants is improved by applying a localizing scheme, and the corresponding approximation error is estimated. A noteworthy case in Hilbert spaces and a numerical test comparing the Hermite-Birkhoff and Lagrange interpolants complete the presentation.

Keywords:

• Cardinal basis function
• Hermite-Birkhoff interpolation
• Interpolation in Banach spaces

Mathematics Subject Classification:

• 65D05
• 46N40