By Adhemar Bultheel, Pablo Gonzalez-Vera, Erik Hendriksen, Olav Njastad

This quantity generalizes the classical idea of orthogonal polynomials at the advanced unit circle or at the actual line to orthogonal rational services whose poles are between a prescribed set of advanced numbers. the 1st half treats the case the place those poles are all outdoor the unit disk or within the reduce part airplane. Classical issues reminiscent of recurrence family members, numerical quadrature, interpolation houses, Favard theorems, convergence, asymptotics, and second difficulties are generalized and taken care of intimately. an analogous themes are mentioned for a unique scenario the place the poles can be found at the unit circle or at the prolonged actual line. within the final bankruptcy, a number of purposes are pointed out together with linear prediction, Pisarenko modeling, lossless inverse scattering, and community synthesis. This thought has many purposes either in theoretical genuine and intricate research, approximation concept, numerical research, method idea, and electric engineering.

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5. J-unitary and J-contractive matrices 41 The following theorem describes a simple matrix from the class By. 4. 50). Proof. This is a classical result that can be found, for example, in Potapov [180, pp. 187-188]. • A matrix in N2x2 that is J-contractive in O and J-unitary will be called J-inner, a terminology used by Dym [77]. The set of J-inner matrix functions is denoted as Bj = By n Ty while we set Bj = By HTy. 4 is a J-inner matrix. Much more on J-unitary matrix functions and the work of Potapov can be found in the V.

Measure. 37), the Blaschke factor of the inner factor U catches all the zeros an off. Inner functions in O have apseudo-meromorphic extension across the boundary dO [73]. This means the following: Because U e Z/^, it is an analytic function in O and therefore (7* e H'p, and thus it is analytic in Oe. Moreover, on the boundary 9O since we have almost everywhere for any inner function \U\2 = 1 or UU* = 1, we can write U = 1/U* on dO. In this way, U has an analytic extension to the whole complex Riemann sphere, where we have to exclude the poles ay, j = 1, 2 , .

Problem P2(a, w) can be understood as the problem of finding the orthogonal projection in H of 0 onto the space V = {/ e H : f(w) = a}. 5. J-unitary and J-contractive matrices We shall consider 2 x 2 matrices 0 whose entries are functions in the Nevanlinna class N: 0 = [ft7] e N2x2. We consider such matrices that are unitary with respect to the indefinite metric . 0 0 -1 = 1 -1. 46) where the superscript H denotes complex conjugate transpose. 5. J-unitary and J-contractive matrices 37 As we did for inner functions in S, we can define a pseudo-meromorphic extension for such a 0-matrix.