Proceedings of the American Mathematical Society

Measures with finite index of determinacy or a mathematical model for Dr. Jekyll and Mr. Hyde

Author(s): Christian Berg; Antonio J. Duran
Journal: Proc. Amer. Math. Soc. 125 (1997), 523-530.
MSC (1991): Primary 42C05, 44A60
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Abstract: In this note measures with finite index of determinacy (i.e. determinate measures $\mu$ for which there exists a polynomial $p$

such that $\vert p\vert ^{2} \mu$ is indeterminate) are characterizated in terms of the operator associated to its Jacobi matrix. Using this characterization, we show that such determinate measures with finite index of determinacy (Jekyll) turn out to be indeterminate (Hyde) when considered as matrices of measures.

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[BD2]: -, When does a discrete differential perturbation of a sequence of orthonormal polynomials belong to $\ell ^{2}$ ?, J. Funct. Anal. 136 (1996), 127-153. CMP 96:08
[D1]: Duran, A.J., A generalization of Favard's theorem for polynomials satisfying a recurrence relation, J. Approx. Theory 74 (1993), 83-109. MR 94k:41008
[D2]: -, On Orthogonal polynomials with respect to a positive definite matrix of measures, Can. J. Math. 47 (1995), 88-112. MR 96c:42047
[DV]: Duran, A.J. and van Assche, W., Orthogonal matrix polynomials and higher order recurrence relations, Linear Algebra Appl. 219 (1995), 261-280. MR 96c:42048
[R]: Riesz, M., Sur le problème des moments. Troisième Note., Arkiv för Mat., astr. och fys. 17 (1923), no. 16.

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 42C05, 44A60

Christian Berg
Affiliation: Matematisk Institut, Københavns Universitet, Universitetsparken 5, DK-2100 Køben- havn Ø, Denmark
Email: berg@math.ku.dk

Antonio J. Duran
Affiliation: Departamento de Análisis Matemático, Universidad de Sevilla, Apdo. 1160. 41080-Sevilla, Spain
Email: duran@cica.es

DOI: 10.1090/S0002-9939-97-03613-7
PII: S 0002-9939(97)03613-7
Received by editor(s): August 29, 1995
Additional Notes: This work has been partially supported by DGICYT ref. PB93-0926.
Communicated by: Palle E. T. Jorgensen
Copyright of article: Copyright 1997, American Mathematical Society

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