Convergence of Rational Interpolants with Preassigned Poles

✍ Scribed by Amiran Ambroladze; Hans Wallin

• Publisher: Elsevier Science
• Year: 1997
• Tongue: English
• Weight: 336 KB
• Volume: 89
• Category: Article
• ISSN: 0021-9045
• DOI: 10.1006/jath.1997.3040

✦ Synopsis

We study the following problem. Given a domain 0 containing infinity, is it possible to choose a sequence of polynomials Q n , n=1, 2, ..., where Q n has degree n, so that the following condition holds: if a function f is analytic in 0 and P n is the polynomial part of the Laurent expansion of Q n f at infinity, then P n ÂQ n converges to f, as n tends to infinity, uniformly on bounded closed subsets of 0? We get a complete solution of this problem when 0 is regular for Dirichlet's problem.

For irregular domains we obtain some results having independent interest but a main problem remains open: is it possible to find such polynomials Q n for some irregular domains 0?

1997 Academic Press and this series converges inside the maximal disk, centered at infinity, where f is analytic, and diverges outside this disk.

This incomplete form of convergence, which is probably the main disadvantage of Taylor series comes from the fact that the partial sums of Taylor article no. AT973040 238