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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

The index of determinacy for measures and the $ l\sp 2$-norm of orthonormal polynomials


Authors: Christian Berg and Antonio J. Duran
Journal: Trans. Amer. Math. Soc. 347 (1995), 2795-2811
MSC: Primary 30E05; Secondary 30D15, 42C05, 44A60
MathSciNet review: 1308001
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Abstract: For determinate measures $ \mu $ having moments of every order we define and study an index of determinacy which checks the stability of determinacy under multiplication by even powers of $ \vert t - z\vert$ for $ z$ a complex number. Using this index of determinacy, we solve the problem of determining for which $ z \in \mathbb{C}$ the sequence $ {(p_n^{(m)}(z))_n}(m \in \mathbb{N})$ belongs to $ {\ell ^2}$, where $ {({p_n})_n}$ is the sequence of orthonormal polynomials associated with the measure $ \mu $.


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Additional Information

DOI: http://dx.doi.org/10.1090/S0002-9947-1995-1308001-9
PII: S 0002-9947(1995)1308001-9
Keywords: Moment problem, index of determinacy, $ {\text{N}}$-extremal measures, orthogonal polynomials
Article copyright: © Copyright 1995 American Mathematical Society