Circular parameters of polynomials orthogonal on several arcs of the unit circle

A. L. Lukashov

Sbornik. Mathematics, Tome 195 (2004) no. 11, pp. 1639-1663

Résumé

The asymptotic behaviour of the circular parameters $(a_n)$ of the polynomials orthogonal on the unit circle with respect to Geronimus measures is analysed. It is shown that only when the harmonic measures of the arcs making up the support of the orthogonality measure are rational do the corresponding parameters form a pseudoperiodic sequence starting from some index (that is, after a suitable rotation of the circle and the corresponding modification of the orthogonality measures they form a periodic sequence). In addition it is demonstrated that if the harmonic measures of these arcs are linearly independent over the field of rational numbers, then the sets of limit points of the sequences of absolute values of the circular parameters $|a_n|$ and of their ratios $(a_{n+k}/a_n)_{n=1}^\infty$ are a closed interval on the real line and a continuum in the complex plane, respectively.

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