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The supports of measures associated with orthogonal polynomials and the spectra of the related self-adjoint operators. (English) Zbl 0731.42021

Summary: An elementary approach is given to prove Blumenthal’s theorem describing the support of measures associated with orthogonal polynomials on the real line in case the recurrence coefficients associated with these polynomials tend to finite limits. Then the known approach using H. Weyl’s theorem on compact perturbations of selfadjoint operators to Blumenthal’s theorem is presented. Finally, using Weyl’s theorem, Geronimus’ result on the support is discussed when the recurrence coefficients with subscripts having the same residue (mod k) have finite limits. Instead of the usual approach of using continued fractions, the Hardy class \(H^ 2\) is used to determine the spectrum of the selfadjoint operator arising in the study of this support.

MSC:

42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis
47B15 Hermitian and normal operators (spectral measures, functional calculus, etc.)
47A10 Spectrum, resolvent
30D55 \(H^p\)-classes (MSC2000)
39A10 Additive difference equations
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