Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle

Details Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle Five-diagonal matrices and zeros of orthogonal polynomials on the unit circle

2002-04-24

arxiv.org/abs/math/0204300v1

Mathematics

Classical Analysis and ODEs

25 pages

Scientific paper

It is shown that monic orthogonal polynomials on the unit circle are the characteristic polynomials of certain five-diagonal matrices depending on the Schur parameters. This result is achieved through the study of orthogonal Laurent polynomials on the unit circle. More precisely, it is a consequence of the five term recurrence relation obtained for these orthogonal Laurent polynomials, and the one to one correspondence established between them and the orthogonal polynomials on the unit circle. As an application, some results relating the behavior of the zeros of orthogonal polynomials and the location of Schur parameters are obtained.

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