Spectral methods for orthogonal rational functions

An operator theoretic approach to orthogonal rational functions on the unit circle with poles in its exterior is presented in this paper. This approach is based on the identification of a suitable matrix representation of the multiplication operator associated with the corresponding orthogonality measure. Two different alternatives are discussed, depending whether we use for the matrix representation the standard basis of orthogonal rational functions, or a new one with poles alternatively located in the exterior and the interior of the unit circle. The corresponding representations are linear fractional transformations with matrix coefficients acting respectively on Hessenberg and five-diagonal unitary matrices. In consequence, the orthogonality measure can be recovered from the spectral measure of an infinite unitary matrix depending uniquely on the poles and the parameters of the recurrence relation for the orthogonal rational functions. Besides, the zeros of the orthogonal and para-orthogonal rational functions are identified as the eigenvalues of matrix linear fractional transformations of finite Hessenberg and fivediagonal matrices. ∗This work was partially realized during a stay of the author at the Norwegian University of Science and Technology financed by Secretaŕıa de Estado de Universidades e Investigación from the Ministry of Education and Science of Spain. The work of the author was also partly supported by a research grant from the Ministry of Education and Science of Spain, project code MTM2005-08648-C02-01, and by Project E-64 of Diputación General de Aragón (Spain).