An extension of the associated rational functions on the unit circle
Deckers, Karl × Cantero, Maria Jose Moral, Leandro Velazquez, Luis #
Journal of Approximation Theory vol:163 issue:4 pages:524-546
A special class of orthogonal rational functions (ORFs) is presented in this paper. Starting with a sequence of ORFs and the corresponding rational functions of the second kind, we define a new sequence as a linear combination of the previous ones, the coefficients of this linear combination being self-reciprocal rational functions. We show that, under very general conditions on the self-reciprocal coefficients, this new sequence satisfies orthogonality conditions as well as a recurrence relation. Further, we identify the Carathéodory function of the corresponding orthogonality measure in terms of such self-reciprocal coefficients.
The new class under study includes the associated rational functions as a particular case. As a consequence of the previous general analysis, we obtain explicit representations for the associated rational functions of arbitrary order, as well as for the related Carathéodory function. Such representations are used to find new properties of the associated rational functions.