Positive functionals and Hessenberg matrices
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- by Jean B. Lasserre and Mihai Putinar PDF
- Proc. Amer. Math. Soc. 147 (2019), 1097-1108 Request permission
Abstract:
Not every positive functional defined on bi-variate polynomials of a prescribed degree bound is represented by the integration against a positive measure. We isolate a couple of conditions filling this gap, either by restricting the class of polynomials to harmonic ones, or imposing the vanishing of a defect indicator. Both criteria offer constructive cubature formulas and they are obtained via well-known matrix analysis techniques involving either the dilation of a contractive matrix to a unitary one or the specific structure of the Hessenberg matrix associated to the multiplier by the underlying complex variable.References
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Additional Information
- Jean B. Lasserre
- Affiliation: LAAS-CNRS and Institute of Mathematics, University of Toulouse, France
- MR Author ID: 110545
- Email: lasserre@laas.fr
- Mihai Putinar
- Affiliation: Department of Mathematics, University of California at Santa Barbara, Santa Barbara, California 93117—and—Department of Mathematics, Newcastle University, Newcastle upon Tyne, United Kingdom
- MR Author ID: 142835
- Email: mputinar@math.ucsb.edu, mihai.putinar@ncl.ac.uk
- Received by editor(s): April 20, 2018
- Received by editor(s) in revised form: May 31, 2018
- Published electronically: November 16, 2018
- Additional Notes: Research of the first author and visit of the second author at LAAS in Toulouse, funded by the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation program (grant agreement ERC-ADG 666981 TAMING)
- Communicated by: Stephan Ramon Garcia
- © Copyright 2018 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 147 (2019), 1097-1108
- MSC (2010): Primary 47B20; Secondary 65D32
- DOI: https://doi.org/10.1090/proc/14266
- MathSciNet review: 3896059