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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

Zero distribution of Müntz extremal polynomials in $ L_{p}\left[ 0,1\right] $


Authors: D. S. Lubinsky and E. B. Saff
Journal: Proc. Amer. Math. Soc. 135 (2007), 427-435
MSC (2000): Primary 41A10, 41A17, 42C99; Secondary 33C45
Published electronically: August 4, 2006
MathSciNet review: 2255289
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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

$\displaystyle \lim_{n\rightarrow \infty }\frac{\lambda _{n}}{n}=\alpha >0, $

then the normalized zero counting measure of $ T_{n,p}$ converges weakly as $ n\rightarrow \infty $ to

$\displaystyle \frac{\alpha }{\pi }\frac{t^{\alpha -1}}{\sqrt{t^{\alpha }\left( 1-t^{\alpha} \right) }}dt, $

while if $ \alpha =0$ or $ \infty $, the limiting measure is a Dirac delta at $ 0$ or $ 1$, respectively.


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Additional Information

D. S. Lubinsky
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160
Email: lubinsky@math.gatech.edu

E. B. Saff
Affiliation: Center for Constructive Approximation, Department of Mathematics, Vanderbilt University, Nashville, Tennessee 37240.
Email: Edward.B.Saff@Vanderbilt.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-06-08694-1
PII: S 0002-9939(06)08694-1
Keywords: Zero distribution, M\"{u}ntz extremal polynomials, M\"{u}ntz orthogonal polynomials
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
Article copyright: © Copyright 2006 American Mathematical Society