Remote Access Proceedings of the American Mathematical Society
Green Open Access

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
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • 1. V.V. Andrievskii and H.P. Blatt, Discrepancy of Signed Measures and Polynomial Approximation, Springer, New York, 2002. MR 1871219 (2002k:30001)
  • 2. P. Borwein and T. Erdelyi, Polynomials and Polynomial Inequalities, Springer, New York, 1995. MR 1367960 (97e:41001)
  • 3. G. V. Milovanovic, D. S. Mitrinovic, and Th. M. Rassias, Topics in Polynomials: Extremal Problems, Inequalities, Zeros, World Scientific, Singapore, 1994. MR 1298187 (95m:30009)
  • 4. A. Pinkus and Z. Ziegler, Interlacing Properties of the Zeros of Error Functions in Best $ L_{p}$ Approximations, J. Approx. Theory, 27(1979), 1-18. MR 0554112 (81c:41063)
  • 5. E. B. Saff and V. Totik, Logarithmic Potentials with External Fields, Springer, New York, 1997. MR 1485778 (99h:31001)
  • 6. E. B. Saff and R. S. Varga, On Lacunary Incomplete Polynomials, Math. Zeitschrift, 177(1981), 297-314. MR 0618197 (83a:41008)
  • 7. H. Stahl and V. Totik, General Orthogonal Polynomials, Cambridge University Press, Cambridge, 1992. MR 1163828 (93d:42029)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 41A10, 41A17, 42C99, 33C45

Retrieve articles in all journals with MSC (2000): 41A10, 41A17, 42C99, 33C45

Additional Information

D. S. Lubinsky
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332-0160

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

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

American Mathematical Society