Orthonormal bases of polynomials in one complex variable

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Let a sequence (Pn) of polynomials in one complex variable satisfy a recurrence relation with length growing more slowly than linearly. It is shown that (Pn) is an orthonormal basis in L2μ for some measure μ on C, if and only if the recurrence is a 3-term relation with special coefficients. The support 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.

Original languageEnglish
Pages (from-to)7-14
Number of pages8
JournalAnalysis Mathematica
Issue number1
StatePublished - 2003


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