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Orthogonal polynomials on several intervals: accumulation points of recurrence coefficients and of zeros. (English) Zbl 1225.33010

This work was presented during the international conference on Orthogonal Polynomials, Special Functions and Applications held at the Katholieke Universiteit Leuven (Belgium), 20–25 July, 2009.
As Franz Peherstorfer died on November 27, 2009, this was his last contribution to an international conference and the paper has been published posthumously.
It is a deep and intricate paper, treating an important problem concerning the accumulation behavior of the recurrence coefficients of the orthonormal polynomials with respect to an absolutely continuous measure on the union of disjoint intervals (with at most a finite number of point measures outside the union).
It would go too far to give the detailed results of this densely written paper in a review and therefore the abstract will be given below.
One wonders how much more Franz could have contributed to the field of orthogonal polynomials and approximation theory, had his untimely dead not occurred: may he rest in peace. So let us quote the author’s abstract:
“Let \(E=\bigcup_{j=1}^ {\ell}\,[a_{2j-1},a_{2j}]\) be the union of \(\ell\) disjoint intervals and set \(\omega(\infty)= (\omega_1(\infty),\dots,\omega_{\ell-1}(\infty))\), where \(\omega_j(\infty)\) is the harmonic measure of \([a_{2j-1},a_{2j}]\) at infinity. Let \(\mu\) be a measure which is absolutely continuous on \(E\), satisfying Szegö’s condition, and with at most a finite number of point measures outside \(E\), and denote by \((P_n)\) and \(({\mathcal Q}_n)\) the orthonormal polynomials and their associated Weyl solution with respect to \(\mu\). We show that the recurrence coefficients have topologically the same convergence behavior as the sequence \((n\omega(\infty))_{n\in\mathbb{N}}\) modulo \(1\). As one of the consequences, there is a homeomorphism from the so-called gaps \(\text{X}_{j=1}^{\ell-1}\,([a_{2j},a_{2j+1}]^{+} \cup [a_{2j},a_{2j+1}]^{-})\) on the Riemann surface for \(y^2=\prod_{j=1}^{2\ell}\,(x-a_{j})\) into the set of accumulation points of the sequence of recurrence coefficients if \(\omega_1(\infty),\dots,\omega_{\ell-1}(\infty),1\) are linearly independent over the rational numbers \(\mathbb{Q}\).
Furthermore, it is shown that the convergence behavior of the sequence or recurrence coefficients and the sequence of zeros of the orthonormal polynomials and Weyl solutions outside the spectrum is topologically the same.
These results are proved by proving corresponding statements for accumulation points of the vector of moments of the diagonal Green’s functions.”

MSC:

33C45 Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
41A50 Best approximation, Chebyshev systems

Biographic References:

Peherstorfer, Franz
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References:

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