Colóquio 

14/03/2002, 17:00 — 18:00 — Anfiteatro Pa1, Pavilhão de Matemática
Israel Gohberg, Tel Aviv University

Orthogonal systems and convolution equations

This talk is about orthogonal polynomials and their generalizations. The polynomials are obtained by orthogonalization of the sequence of power functions on the unit circle in respect to weighted inner products. The impetus for these results was a theorem of M. G. Krein and its generalization by M. G. Krein and H. Langer. The story started with a theorem of Szego which proved that this type of orthogonal polynomial in the case of a positive weight has all zeros inside the unit circle. M. G. Krein extended this result for the case when the weight is changing signs. M. G. Krein and H. Langer obtained the continuous analog of the latter result. We will present all these theorems together with the one step version. The inverse theorem will also be considered. The next topic will be the matrix orthogonal polynomials and Krein's theorem. Also will be presented other generalizations based on orthogonalization in modules. The theory of Toeplitz and Wiener-Hopf operators plays an important role in these considerations. The talk is based on recent results of A. Ben-Artzi, R. Ellis, I. Gohberg, D. Lay, and L. Lerer, and it is planned for a wide audience.