Connection between orthogonal polynomials on the unit circle and bounded interval
The Chebyshev polynomials \(T_n(x)=\cos(n\arccos x)\) can be written \(T_n(x)=\frac {z^n+z^{-n}}2\) for \(x =\frac {z+z^{-1}}{2}\,.\) The authors consider this type of transformation between measures on a bounded interval and the unit circle due to Szegő in detail. They relate the sequences of orthogonal polynomials with respect to both measures and they also relate the coefficients of the three-term recurrence relation with the Schur parameters. When the measures belong to the Szegő class, they study the asymptotic behavior of the orthogonal polynomials on the interval, outside the support of the measure, as well as inside. They also transform the generalized polynomials and they study the orthogonality properties of the new polynomials, obtaining new interesting results. Finally they solve two inverse problems connected with the transformation studied.
scientific article; zbMATH DE number 2136625