Small values of polynomials and potentials with $L_p$ normalization
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- by D. S. Lubinsky PDF
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Abstract:
For a polynomial $P$ of degree $\leq n$, normalized by the condition \[ \frac 1{2\pi }\int _0^{2\pi }\mid P(re^{i\theta })\mid ^pd\theta =1, \] we show that $E(P;r;\varepsilon ):=\{z:\mid z\mid \leq r,\mid P(z)\mid \leq \varepsilon ^n\}$ has $cap$ at most $r\varepsilon \kappa _{np}$, where $\kappa _{np}\leq 2$ is explicitly given and sharp for each $n,r$. Similar estimates are given for other normalizations, such as $p=0$, and for planar measure, and for generalized polynomials and potentials, thereby extending work of Cuyt, Driver and the author for $p=\infty$. The relation to Remez inequalities is briefly discussed.References
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Additional Information
- D. S. Lubinsky
- Affiliation: Department of Mathematics, Witwatersrand University, Wits 2050, South Africa
- MR Author ID: 116460
- ORCID: 0000-0002-0473-4242
- Email: 036dsl@cosmos.wits.ac.za
- Received by editor(s): July 17, 1996
- Received by editor(s) in revised form: June 2, 1997
- Communicated by: J. Marshall Ash
- © Copyright 1999 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 127 (1999), 529-536
- MSC (1991): Primary 30C10, 31A15; Secondary 41A17, 41A44, 30C85
- DOI: https://doi.org/10.1090/S0002-9939-99-04549-9
- MathSciNet review: 1468199