Orthonormal bases of polynomials in one complex variable
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 $L^2_{\mu}$ 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.
 Publication:

arXiv Mathematics eprints
 Pub Date:
 November 2000
 arXiv:
 arXiv:math/0011240
 Bibcode:
 2000math.....11240C
 Keywords:

 Functional Analysis;
 41A10;
 47B15
 EPrint:
 5 pages