A Vector Generalisation of de Montessusʹ Theorem for the Case of Polar Singularities on the Boundary Original Research Article

In the context of vector Padé approximants we present an extension of de Montessusʹ 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.