2 research outputs found
Orthogonal rational functions and rational modifications of a measure on the unit circle
AbstractIn this paper we present formulas expressing the orthogonal rational functions associated with a rational modification of a positive bounded Borel measure on the unit circle, in terms of the orthogonal rational functions associated with the initial measure. These orthogonal rational functions are assumed to be analytic inside the closed unit disc, but the extension to the case of orthogonal rational functions analytic outside the open unit disc is easily made. As an application we obtain explicit expressions for the orthogonal rational functions associated with a rational modification of the Lebesgue measure
An extension of the associated rational functions on the unit circle
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
Caratheodory 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 Caratheodory function. Such representations
are used to find new properties of the associated rational functions.Comment: 27 page