Paatero’s $V(k)$ space and a claim by Pinchuk
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- by Valentin V. Andreev, Miron B. Bekker and Joseph A. Cima
- Proc. Amer. Math. Soc. 150 (2022), 1711-1717
- DOI: https://doi.org/10.1090/proc/15790
- Published electronically: January 26, 2022
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Abstract:
In this article we obtain a factorization theorem for the functions in Paatero’s $V(k)$ space. We bring attention to a significant result of Pinchuk which unfortunately is false. This result relates measures associated to functions in $V(k)$ and an integral representation theorem for such functions. We prove necessary and sufficient conditions for a wide class of functions (in particular, the polynomials) to belong to the Paatero class based on the geometry of their critical points, and obtain explicit representation of the measures associated to a wide class of such polynomials that includes the Suffridge polynomials.References
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Bibliographic Information
- Valentin V. Andreev
- Affiliation: Department of Mathematics, Lamar University, Beaumont, Texas 77710
- MR Author ID: 25995
- ORCID: 0000-0002-9929-7048
- Email: vvandreev8@gmail.com
- Miron B. Bekker
- Affiliation: Department of Mathematics, the University of Pittsburgh at Johnstown, 450 Schoolhouse, Johnstown, Pennsylvania 15904
- MR Author ID: 260633
- Email: bekker@pitt.edu
- Joseph A. Cima
- Affiliation: Department of Mathematics, the University of North Carolina at Chapel Hill, CB 3250, 329 Phillips Hall, Chapel Hill, North Carolina 27599
- MR Author ID: 49485
- ORCID: 0000-0001-8740-2579
- Email: cima@email.unc.edu
- Received by editor(s): May 12, 2021
- Received by editor(s) in revised form: July 29, 2021
- Published electronically: January 26, 2022
- Communicated by: Javad Mashreghi
- © Copyright 2022 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 150 (2022), 1711-1717
- MSC (2020): Primary 30C45, 30C15, 30H10
- DOI: https://doi.org/10.1090/proc/15790
- MathSciNet review: 4375757