Multipoint Schur's algorithm, rational orthogonal functions, asymptotic properties and Schur rational approximation

In a paper by Khrushchev, the connections between the Schur algorithm, the Wall's continued fractions and the orthogonal polynomials are revisited and used to establish some nice convergence properties of the sequence of Schur functions associated with a Schur function. In this report, we generalize some of Krushchev's results to the case of a multipoint Schur algorithm, that is a Schur algorithm where all the interpolation points are not taken in 0 but anywhere in the open unit disk. To this end, orthogonal rational functions and a recent generalization of Geronimus theorem are used. Then, we consider the problem of approximating a Schur function by a rational function which is also Schur. This problem of approximation is very important for the synthesis and identification of passive systems. We prove that all strictly Schur rational function of degree $n$ can be written as the $2n$-th convergent of the Schur algorithm if the interpolation points are correctly chosen. This leads to a parametrization using the multipoint Schur algorithm. Some examples are computed by an $L^2$ norm optimization process and the results are validated by comparison with the unconstrained $L^2$ rational approximation

On discrete spectra of bergman-toeplitz operators with harmonic symbols

In the present article, we study the discrete spectrum of certain bounded Toeplitz operators with harmonic symbol on a Bergman space. Using the methods of classical perturbaton theory and recent results by Borichev-Golinskii-Kupin and Favorov-Golinskii, we obtain a quantitative result on the distribution of the discrete spectrum of the operator in the unbounded (outer) component of its Fredholm set

A VERSION OF WATSON LEMMA FOR LAPLACE INTEGRALS AND SOME APPLICATIONS

We obtain the asymptotics of a Laplace-type integral for a function behaving "logarithmically" near the critical point. Actually, the result is proved for a larger class of ``slowly oscillating'' functions satisfying some mild regularity conditions

ON COMPLEX PERTURBATIONS OF INFINITE BAND SCHRÖDINGER OPERATORS

Let H_0 = −d^2/dx^2 + V_0 be an infinite band Schrödinger operator on L^2(R) with a real-valued potential V_0 ∈ L^\infty (R). We study its complex perturbation H = H_0+V , defined in the form sense, and obtain the Lieb-Thirring type inequalities for the rate of convergence of the discrete spectrum of H to the joint essential spectrum. The assumptions on V vary depending on the sign of Re V

A Blaschke-type condition for analytic functions on finitely connected domains. Applications to complex perturbations of a finite-band selfadjoint operator.

This is a sequel of the article by Borichev-Golinskii-Kupin [2009], where the authors obtain Blaschke-type conditions for special classes of analytic functions in the unit disk which satisfy certain growth hypotheses. These results were applied to get Lieb-Thirring inequalities for complex compact perturbations of a selfadjoint operator with a simply connected resolvent set. The first result of the present paper is an appropriate local version of the Blaschke-type condition from Borichev-Golinskii-Kupin [2009]. We apply it to obtain a similar condition for an analytic function in a finitely connected domain of a special type. Such condition is by and large the same as a Lieb-Thirring type inequality for complex compact perturbations of a selfadjoint operator with a finite-band spectrum. A particular case of this result is the Lieb--Thirring inequality for a selfadjoint perturbation of the Schatten class of a periodic (or a finite-band) Jacobi matrix. The latter result seems to be new in such generality even in this framework

Itô diffusions, modified capacity and harmonic measure. Applications to Schrödinger operators.

We observe that some special Itô diffusions are related to scattering properties of a Schrödinger operator on R^d, d>1. We introduce Feynman-Kac type formulae for these stochastic processes which lead us to results on the preservation of the a.c. spectrum of the Schrödinger operator. To better understand the analytic properties of the processes, we construct and study a special version of the potential theory. The modified capacity and harmonic measure play an important role in these considerations. Various applications to Schrödinger operators are also given. For example, we relate the presence of the absolutely continuous spectrum to the geometric properties of the support of the potential

A REMARK ON ANALYTIC FREDHOLM ALTERNATIVE

We apply a recent result of Borichev-Golinskii-Kupin on the Blaschke-type conditions for zeros of analytic functions on the complex plane with a cut along the positive semi-axis to the problem of the eigenvalues distribution of the Fredholm-type analytic operator-valued functions