Continuous analogs of polynomials orthogonal on the unit circle. Krein systems

By Sergey A. Denisov. 2009

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Preface In the recent years, the theory of orthogonal polynomials on the real line (OPRL) and on the unit circle (OPUC) enjoyed the considerable development. In these lecture notes, we will explain how to construct the continuous analogs of polynomials orthogonal on the unit circle. It is possible to built a theory which is as rigorous and complete as the theory for OPUC. Spectral theories of one-dimensional Dirac and Schrodinger operators can then be viewed in the framework of this theory which establishes a solid link between an approximation theory and quantum mechanics. The theory is based on the ideas suggested by M.G. Krein. They were developed later by various authors, especially from Krein's school. In the meantime, new results were obtained for OPUC and OPRL and that was a motivation for us to try to understand their continuous analogs. We also try to give systematic exposition of the theory but have to refer to the literature once in a while. Also, these notes do not cover some aspects of the theory (e.g. continuation problems for G r classes, regularity of coefficients, etc.) but we give necessary references. In general, our objective is to give only basics of the theory by presenting complete proofs and filling various gaps present in the current literature. As a prerequisite for reading these notes, we assume that a reader is familiar with main facts from the OPUC theory (see, e.g. [72, 26, 66]). The knowledge of spectral theory for Schrödinger and Dirac operators might also be very helpful. What is not covered and what is new? We didn't include the following subjects that are related to our topic: solution to the continuous analogs of Schur and Caratheodory-Toeplitz problems [45, 46]. We also do not discuss matrix-valued version of the theory. For the recent progress on more specific questions (such as continuous analog of Szeg˝ o case, Rakhmanov's Theorem, etc.) we suggest the reader to consult the journal publications, e.g. Also, we will deal with rather regular classes of coefficients (not worse than L 2 loc (R +)) but the general case can also be treated in the framework of different differential operators (see, e.g., [52, 53, 6]). In these notes, we present quite a few new results. That includes: approximation of continuous orthogonal system by the sequence of the discrete ones (Section 8), distribution of zeroes (Section 9), new criteria for A(r) ∈ L …

Subfields: Applied Mathematics, Mathematical Physics, Statistical and Nonlinear Physics

View Paper: https://www.semanticscholar.org/paper/7ef4ce7e04886880a20ff523a8ec1bf7d30183a8

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