An optimal approximation problem for free polynomials
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- by Palak Arora, Meric Augat, Michael T. Jury and Meredith Sargent;
- Proc. Amer. Math. Soc. 152 (2024), 455-470
- DOI: https://doi.org/10.1090/proc/16474
- Published electronically: November 17, 2023
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Abstract:
Motivated by recent work on optimal approximation by polynomials in the unit disk, we consider the following noncommutative approximation problem: for a polynomial $f$ in $d$ freely noncommuting arguments, find a free polynomial $p_n$, of degree at most $n$, to minimize $c_n ≔\|p_nf-1\|^2$. (Here the norm is the $\ell ^2$ norm on coefficients.) We show that $c_n\to 0$ if and only if $f$ is nonsingular in a certain nc domain (the row ball), and prove quantitative bounds. As an application, we obtain a new proof of the characterization of polynomials cyclic for the $d$-shift.References
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Bibliographic Information
- Palak Arora
- Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611
- ORCID: 0009-0005-1418-5725
- Email: pa8@williams.edu
- Meric Augat
- Affiliation: Department of Mathematics, University of South Florida, Tampa, Florida 33620
- MR Author ID: 1171184
- ORCID: 0000-0003-3125-1155
- Email: mla026@bucknell.edu
- Michael T. Jury
- Affiliation: Department of Mathematics, University of Florida, Gainesville, Florida 32611
- MR Author ID: 742791
- Email: mjury@ufl.edu
- Meredith Sargent
- Affiliation: Department of Mathematics, University of Manitoba, Winnipeg, Manitoba R3T 2N2, Canada
- MR Author ID: 1278047
- ORCID: 0000-0002-7505-0841
- Email: meredithsargent@gmail.com
- Received by editor(s): October 7, 2022
- Received by editor(s) in revised form: March 1, 2023
- Published electronically: November 17, 2023
- Additional Notes: The first author’s research was supported by NSF grant DMS-2154494.
The second author’s research was supported by NSF grant DMS-2155033.
The third author’s research was supported by NSF grant DMS-2154494.
The fourth author was supported by the Pacific Institute for the Mathematical Sciences (PIMS). The research and findings may not reflect those of the Institute. - Communicated by: Javad Mashreghi
- © Copyright 2023 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 152 (2024), 455-470
- MSC (2020): Primary 16S38, 46L52, 47A16; Secondary 47A13
- DOI: https://doi.org/10.1090/proc/16474
- MathSciNet review: 4683831