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Let R be the space of rational functions with poles among f k ; 1= k g 1 k=0 with 0 = 0 and j k j < 1, k 1. We consider a sequence fR n g 1 n=0 of nested subspaces with 1 n=0 R n = R. First we recall from part I how to nd orthogonal bases for R for a positive measure on the unit circle. These are used in the construction of interpolatory quadrature rules for integrals with respect to a complex measure on the unit circle. Integration for the (2n + 1)-point rule is exact for all f 2 R n. Also their convergence is discussed as n ! 1. Finally we discuss the convergence of multipoint rational approximants to the Riesz-Herglotz transform associated with such a complex measure.

ORTHOGONAL RATIONAL FUNCTIONS AND INTERPOLATORY PRODUCT RULES ON THE UNIT CIRCLE: III: CONVERGENCE OF GENERAL SEQUENCES