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The dyadic distribution and its orthogonal polynomials. (English) Zbl 1204.42039

Summary: 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 \(2k-1\) 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 Szegő’s classical theorem.

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
44A60 Moment problems
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