Connection between orthogonal polynomials on the unit circle and bounded interval

Abstract

In this paper we establish the connection between measures on a bounded interval and on the unit circle by a transformation related with the Szegő classical transformation. We transform a measure on the interval [ - 1 , 1 ] into a measure on [ 0 , 2 π ] in the same form in which the orthogonality measure of the Chebyshev polynomials of fourth kind becomes the Lebesgue measure. We relate the sequences of orthogonal polynomials with respect to both measures and we also relate the coefficients of the three-term recurrence relation with the Schur parameters. When the measures belong to the Szegő class, we study the asymptotic behavior of the orthogonal polynomials on the interval, outside the support of the measure, as well as inside. We also transform the generalized polynomials and we study the orthogonality properties of the new polynomials, obtaining new interesting results, and finally we solve two inverse problems connected with the transformation studied.