Complex Mean-Value Interpolation and Approximation of Holomorphic Functions

✍ Scribed by Lars Filipsson

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 506 KB
• Volume: 91
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1996.3096

✦ Synopsis

We show that complex mean-value interpolation, a generalization of Lagrange Hermite interpolation, may be defined in any domain that is C-convex, whereas the original definition required ordinary, real convexity. We also show that C-convex domains are the natural ones in which to perform mean-value interpolation, in the sense that any Runge domain which admits mean-value interpolation must in fact be C-convex. Finally, we obtain an integral formula for the error and give some applications concerning approximation of holomorphic functions.

1997 Academic Press

This polynomial of degree k&m has the property of matching certain canonical mean-values of the function f.

These one variable methods were extended to R n by Goodman [14] and their extensions are sometimes referred to as the scale of mean-value interpolations. As special cases there appear the analogues in R n of article no. AT963096 244