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April 10, 2015 By

The dyadic distribution and its orthogonal polynomials

C. R. Math. Rep. Acad. Sci. Canada Vol. 29 (2) 2007, pp. 52–60

June 30, 2007

Peter C. Gibson, Department of Mathematics & Statistics, York University, Toronto, ON M3J 1P3; email: pcgibson@yorku.ca

Abstract/Résumé:

An open inverse problem that generalizes the classical moment problem is to construct all probability distributions on the real line whose sequence of orthogonal polynomials includes a prescribed subsequence. We have recently solved this problem for a class of subsequences that arise naturally in the context of iterative quadrature schemes, thereby making it possible to construct previously unknown distributions whose orthogonal polynomials have exotic properties. The results are illustrated here by an example: we explicitly construct a distribution on the interval \([-1,1]\), such that for every \(k\geq 1\), its degree \(2^k-1\) orthogonal polynomial divides that of degree \(2^{k+1}-1\), and the zeros of these are equally spaced. Equal spacing of the zeros contrasts starkly with the generic asymptotic behaviour predicted by Szegö’s classical theorem.

Un problème inverse qui reste ouvert et qui généralise le problème classique des moments est de construire toutes les lois de probabilité sur la droite réelle dont la suite des polynômes orthogonaux associée comprend une sous-suite prescrite. On a récemment résolu le problème pour une classe de sous-suites qui provient naturellement des schémas de quadrature iteratifs, ce qui rend possible la construction de lois de probabilités nouvelles dont les polynômes orthogonaux ont des propriétes exotiques. Les résultats sont illustrés ici par un exemple: on construit explicitement une loi sur l’intervalle \([-1,1]\), tel que pour tout \(k\geq 1\), son polynôme orthogonal de degré \(2^k-1\) divise celui de degré \(2^{k+1}-1\), et les zéros de ceux-ci sont également distribués. La distribution égale des zéros se differencie de la distribution asymptotique générique prédite par le théorème classique de Szegö.

Keywords: location of zeros, moment problems, orthogonal polynomials
AMS Subject Classification: Orthogonal functions and polynomials; general theory 42C05

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