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A de Montessus theorem for vector valued rational interpolants. (English) Zbl 0554.41025
Rational approximation and interpolation, Proc. Conf., Tampa/Fla. 1983, Lect. Notes Math. 1105, 227-242 (1984).
[For the entire collection see Zbl 0544.00011.]
The theorem of R. de Montessus de Ballore is extended (by using complex variable methods) to the cases of simultaneous Padé approximation, vector-valued Padé approximation and vector-valued rational interpolation. Let the functions $$f_ 1(z)$$, $$f_ 2(z),...,f_ d(z)$$ be analytic at the origin and meromorphic in a disc $${\mathcal D}_ R:=\{z:| z| <R\}$$ and obey a condition (called polewise independence) on the number and multiplicity of their zeros within $${\mathcal D}_ R$$. We prove convergence of simultaneous Padé approximants to such functions on any compact subset of $${\mathcal D}_ R$$ containing no poles of $$f_ 1(z)$$, $$f_ 2(2),...,f_ d(z)$$, and we derive the rate of convergence of these approximants. The previous results are generalized to the case of vector-valued rational interpolation using the techniques of the second author (1972) and Warner (1976).

MSC:
 41A21 Padé approximation 41A20 Approximation by rational functions
Zbl 0544.00011