Zero distribution of Müntz extremal polynomials in $L_p[0,1]$
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- by D. S. Lubinsky and E. B. Saff PDF
- Proc. Amer. Math. Soc. 135 (2007), 427-435 Request permission
Abstract:
Let $\left \{ \lambda _{j}\right \} _{j=0}^{\infty }$ be a sequence of distinct positive numbers. Let $1\leq p\leq \infty$ and let $T_{n,p}=T_{n,p}\left \{ \lambda _{0},\lambda _{1},\lambda _{2},\dots ,\lambda _{n}\right \} \left ( x\right )$ denote the $L_{p}$ extremal Müntz polynomial in $\left [ 0,1\right ]$ with exponents $\lambda _{0},\lambda _{1},\lambda _{2},\dots ,\lambda _{n}$. We investigate the zero distribution of $\left \{ T_{n,p}\right \} _{n=1}^{\infty }$. In particular, we show that if \begin{equation*} \lim _{n\rightarrow \infty }\frac {\lambda _{n}}{n}=\alpha >0, \end{equation*} then the normalized zero counting measure of $T_{n,p}$ converges weakly as $n\rightarrow \infty$ to \begin{equation*} \frac {\alpha }{\pi }\frac {t^{\alpha -1}}{\sqrt {t^{\alpha }\left ( 1-t^{\alpha } \right ) }}dt, \end{equation*} while if $\alpha =0$ or $\infty$, the limiting measure is a Dirac delta at $0$ or $1$, respectively.References
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Additional Information
- D. S. Lubinsky
- Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160
- MR Author ID: 116460
- ORCID: 0000-0002-0473-4242
- Email: lubinsky@math.gatech.edu
- E. B. Saff
- Affiliation: Center for Constructive Approximation, Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240.
- MR Author ID: 152845
- Email: Edward.B.Saff@Vanderbilt.edu
- Received by editor(s): August 29, 2005
- Published electronically: August 4, 2006
- Additional Notes: The research of the first author was supported by NSF grant DMS0400446. The research of the second author was supported by NSF grant DMS0532154
- Communicated by: Jonathan M. Borwein
- © Copyright 2006 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 135 (2007), 427-435
- MSC (2000): Primary 41A10, 41A17, 42C99; Secondary 33C45
- DOI: https://doi.org/10.1090/S0002-9939-06-08694-1
- MathSciNet review: 2255289