Computational Methods and Function Theory 9 (2009), No. 2, 391--405
Copyright Heldermann Verlag 2009
Line Baribeau
Line.Baribeau@mat.ulaval.ca , Département de mathématiques et statistique, Université Laval, Québec (Québec), G1K 7P4, Canada.
Patrice Rivard
patrice.rivard.1@ulaval.ca , Département de mathématiques et statistique, Université Laval, Québec (Québec), G1K 7P4, Canada.
Elias Wegert
wegert@math.tu-freiberg.de , Institute of Applied Analysis, Tech Univ Bergakademie Freiberg, D-09596 Freiberg, Germany.
Starting from the notion of the complex pseudo-hyperbolic distance and the hyperbolic difference quotient introduced by A. F. Beardon and D. Minda in [1], we define hyperbolic divided differences for unimodularly bounded holomorphic functions in the complex unit disk and investigate their mapping properties. In particular, we show that they operate on Blaschke products in the same way as the ordinary divided differences act on polynomials. As a simple corollary we obtain a multi-point Schwarz-Pick Lemma. Using these concepts we investigate the classical interpolation problem of Pick and Nevanlinna and reformulate the Nevanlinna-Schur algorithm in terms of hyperbolic divided differences. This leads to a scheme that (formally) coincides with Newton's algorithm for polynomial interpolation.
- [1] A. F. Beardon and D. Minda, A multi-point Schwarz-Pick lemma, J. Anal. Math. 92 (2004), 81--104.
Keywords: Hyperbolic geometry, divided differences, Nevanlinna-Pick interpolation, Newton interpolation, Schwarz Lemma.
MSC 2000: Primary 30C80; Secondary 30E05, 30F45.
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