A note on vector-valued rational interpolation

Graves-Morris (see [P.R. Graves-Morris, Vector valued interpolants I, Numer. Math. 42 (1983) 331-348 and P.R. Graves-Morris and C.D. Jenkins, Vector valued rational interpolants III, Constr. Approx. 2 (1986) 263-289]) defined the generalized inverse rational interpolants (GIRIs) in the form of R(x)=N(x)/D(x) with the divisibility condition D(x)∣∥N(x)∥2, and proved the Uniqueness Theorem for GIRIs. However, this condition is found not necessary in some cases. In this paper, we remove this divisibility condition, define the extended generalized inverse rational interpolants (EGIRIs) and establish the Uniqueness Theorem for EGIRIs. One can see that the Uniqueness Theorem for GIRIs is the special case of the one for EGIRIs.