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Orthonormal bases of polynomials in one complex variable

D. P. L. Castrigiano·W. Klopfer
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Abstract

Let a sequence $(P_n)$ of polynomials in one complex variable satisfy a recurre ce relation with length growing slowlier than linearly. It is shown that $(P_n) $ is an orthonormal basis in $L^2_μ$ for some measure $μ$ on $\C$, if and o ly if the recurrence is a $3-$term relation with special coefficients. The supp rt of $μ$ lies on a straight line. This result is achieved by the analysis of a formally normal irreducible Hessenberg operator with only finitely many nonzero entries in every row. It generalizes the classical Favard's Theorem and the Representation Theorem.

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D. P. L. Castrigiano, W. Klopfer. 2000-11-28. Orthonormal bases of polynomials in one complex variable. https://arxiv.org/abs/math/0011240

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