## Final Activity Report Summary - ORTHO-POLYNOMIALS (Orthogonal Polynomials in Spectral Theory)

Orthogonal polynomials (OPs), being one of the most classical mathematical objects, are still in a constant use and of great importance in theory as well as in applications. The three-term recurrence relation for OPs on the real line produces a Jacobi matrix, which is a one-dimensional difference Schrodinger operator. In this way all key properties of orthogonal polynomials transform to spectral properties of the Jacobi matrix. The best known deep classical result of this sort is the Szego theorem, which characterizes the asymptotic behaviour of the Jacobi matrix coefficients and the entries of the generalised eigenvectors under the assumption that their spectral set is an interval and the spectrum has a non-vanishing, absolutely continuous component, satisfying the so-called Szego condition.

One of the main goals of the project was to find general theories for OPs going beyond the Szego class in the stream of three very important results of the last decade, namely the Peherstorfer-Yuditskii (2001), Killip-Simon (2003) and Denisov-Rakhmanov (2004) theorems.

A very essential feature of these results was the presence of an infinite set of eigenvalues of the Jacobi matrix that accumulated to the end of the abovementioned spectral interval, i.e. to the support of the pure point part of the orthogonality measure. In this direction, the most essential result of the project was a counterpart of the Pehersorfer-Yuditskii theory for OPs on the unit circle. More precisely, the orthogonality measure could have an arbitrary component inside the disk, was of the Szego class on the unit circle and was supported on a Blaschke set outside of the disk. It should be mentioned that in the real line case eigenvalues, being on the real line, accumulated to the essential spectrum in a very simple way, radially to the end points of the interval. In the unit circle case the accumulation set was the entire circumference, requiring a completely new approach of proof. We suggested several ones, with the most successful dealing with the Koosis theorem.

Among the applications, a development of the scattering theory on a completely new level was proposed. The project produced a very good background for this goal, in particular a complete description of almost periodic CMV matrices with homogeneous spectrum set; for instance, a system of several intervals or an arbitrary standard Cantor set of a positive Lebesgue measure was given. Generally, based on the project results, a new programme called ‘Orthogonal polynomials and scattering theory’ was submitted and received the support of the Austrian Science Funds (FWF).

Other most essential results were:

1. a proof of the limit periodic property of a Jacobi matrix associated with eigenmeasures of some Ruelle operators, particularly with the balanced measure, on the Julia set of a sufficiently expanding polynomial, i.e. for a polynomial whose critical values exceeded a certain absolute constant;

2. asymptotics of the best polynomial approximation of |x|^p and the best Laurent polynomial approximation of sgn(x) on two symmetric intervals were found including exact constants for the approximation error, i.e. related to Bernstein-kind problems in Approximation Theory, as well as for the best approximation functions under certain renormalisations.

The work was performed in a strong collaboration with the Michigan State University (MSU) analysis group, i.e. with Nazarov-Volberg, as well as researches of other institutions. In the present period several research groups, such as Damanik-Killip-Simon-Zlatos, Denisov-Kupin and Remling, had very important results related to the objectives of the project. Thus all main conjectures dealing with the extension of the Szego theory were completely proved. By the well known close connections of orthogonal polynomials to integration formulas or to the iterative solution of large sparse linear systems of equations, the above results had also immediate applications to this field.

One of the main goals of the project was to find general theories for OPs going beyond the Szego class in the stream of three very important results of the last decade, namely the Peherstorfer-Yuditskii (2001), Killip-Simon (2003) and Denisov-Rakhmanov (2004) theorems.

A very essential feature of these results was the presence of an infinite set of eigenvalues of the Jacobi matrix that accumulated to the end of the abovementioned spectral interval, i.e. to the support of the pure point part of the orthogonality measure. In this direction, the most essential result of the project was a counterpart of the Pehersorfer-Yuditskii theory for OPs on the unit circle. More precisely, the orthogonality measure could have an arbitrary component inside the disk, was of the Szego class on the unit circle and was supported on a Blaschke set outside of the disk. It should be mentioned that in the real line case eigenvalues, being on the real line, accumulated to the essential spectrum in a very simple way, radially to the end points of the interval. In the unit circle case the accumulation set was the entire circumference, requiring a completely new approach of proof. We suggested several ones, with the most successful dealing with the Koosis theorem.

Among the applications, a development of the scattering theory on a completely new level was proposed. The project produced a very good background for this goal, in particular a complete description of almost periodic CMV matrices with homogeneous spectrum set; for instance, a system of several intervals or an arbitrary standard Cantor set of a positive Lebesgue measure was given. Generally, based on the project results, a new programme called ‘Orthogonal polynomials and scattering theory’ was submitted and received the support of the Austrian Science Funds (FWF).

Other most essential results were:

1. a proof of the limit periodic property of a Jacobi matrix associated with eigenmeasures of some Ruelle operators, particularly with the balanced measure, on the Julia set of a sufficiently expanding polynomial, i.e. for a polynomial whose critical values exceeded a certain absolute constant;

2. asymptotics of the best polynomial approximation of |x|^p and the best Laurent polynomial approximation of sgn(x) on two symmetric intervals were found including exact constants for the approximation error, i.e. related to Bernstein-kind problems in Approximation Theory, as well as for the best approximation functions under certain renormalisations.

The work was performed in a strong collaboration with the Michigan State University (MSU) analysis group, i.e. with Nazarov-Volberg, as well as researches of other institutions. In the present period several research groups, such as Damanik-Killip-Simon-Zlatos, Denisov-Kupin and Remling, had very important results related to the objectives of the project. Thus all main conjectures dealing with the extension of the Szego theory were completely proved. By the well known close connections of orthogonal polynomials to integration formulas or to the iterative solution of large sparse linear systems of equations, the above results had also immediate applications to this field.