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OPUC on one foot. (English) Zbl 1108.42005
The theory of orthogonal polynomials on the unit circle (OPUC) invented by G. Szegő in his celebrated 1920–1921 paper has drawn much attention lately, in particular, due to a major development (the discovery of the CMV matrices). Motivated by the dearth of review literature and by the opportunity to use Schrödinger operator techniques in a new setting, the author published two volumes on this issue in 2005.
In the present expository note the way to learn about the subject in less than 1,100 pages is suggested. The author starts out with the basics of the theory (the definition of OPUC, the Szegő recurrences, both in scalar and matrix forms, the Christoffel–Darboux formula, the completeness of OPUC in the corresponding \(L^2_\mu\) space on the unit circle) and then goes over to the contribution of S. Verblunsky and Ya. L. Geronimus to the subject. The focus here is made upon Verblunsky’s “one-one” theorem and the link between OPUC and contractive holomorphic functions in the unit disk (Schur functions) due to Geronimus.
The Bernstein–Szegő approximations and zeros of OPUC are discussed in Section 4. The central Section 5 provides a brief overview of the CMV matrices as a suitable matrix representation for multiplication by \(z\) in \(L^2_\mu\) in the “right” basis.
The rest of the paper concerns Szegő’s and Baxter’s theorems, Khrushchev’s formula, transfer matrices and Weyl solutions, Rakhmanov’s theorem, exponential decay of Verblunsky coefficients and periodic OPUC.

MSC:
42C10 Fourier series in special orthogonal functions (Legendre polynomials, Walsh functions, etc.)
30E05 Moment problems and interpolation problems in the complex plane
42A70 Trigonometric moment problems in one variable harmonic analysis
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