A Vector Generalisation of de Montessus' Theorem for the Case of Polar Singularities on the Boundary

Author: Roberts D.E.

Source: Journal of Approximation Theory, Volume 116, Number 1, May 2002 , pp. 141-168(28)

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Abstract:

In the context of vector Padé approximants we present an extension of de Montessus' theorem to vector-valued meromorphic functions with poles on the boundary of interest—thus strengthening previous results for these approximants. The proofs are framed in Clifford algebras which provide a natural language for discussing vector rational approximants. We also present results for the asymptotic behaviour of the constituent parts of the Clifford denominator—namely, its scalar and bivector parts. In particular, for the case of vector-valued rational functions in which the principal parts are orthogonal to each other for different poles, we demonstrate that the rate of convergence is doubled for the scalar part of the denominator. Finally, we derive consequences of the convergence theorems for the approximation of poles, using either the complete Clifford denominator or its scalar part. © 2002 Elsevier Science (USA).

Language: English

Document Type: Research article

Affiliations: Department of Mathematics, Napier University, 10 Colinton Road, Edinburgh, EH10 5DT, Scotland

Publication date: 2002-05-01