SZEGO and PARA-ORTHOGONAL POLYNOMIALS on THE REAL LINE: ZEROS and CANONICAL SPECTRAL TRANSFORMATIONS

Abstract

We study polynomials which satisfy the same recurrence relation as the Szego polynomials, however, with the restriction that the (reflection) coefficients in the recurrence are larger than one in modulus. Para-orthogonal polynomials that follow from these Szego polynomials are also considered. With positive values for the reflection coefficients, zeros of the Szego polynomials, para-orthogonal polynomials and associated quadrature rules are also studied. Finally, again with positive values for the reflection coefficients, interlacing properties of the Szego polynomials and polynomials arising from canonical spectral transformations are obtained.