Orthonormal bases of polynomials in one complex variable

Details Orthonormal bases of polynomials in one complex variable Orthonormal bases of polynomials in one complex variable

2000-11-28

arxiv.org/abs/math/0011240v1

Mathematics

Functional Analysis

5 pages

Scientific paper

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 $L^2_{\mu}$ for some measure $\mu$ on $\C$, if and o ly if the recurrence is a $3-$term relation with special coefficients. The supp rt of $\mu$ 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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