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On the Smallest and Largest Zeros of Müntz-Legendre Polynomials

  • Úlfar F. Stefánsson (a1)
Abstract

Müntz-Legendre polynomials Ln(Λ x) associated with a sequence Λ = {λk} are obtained by orthogonalizing the system (xλ0, xλ1 , xλ2, … ) in L2[0, 1] with respect to the Legendre weight. If the λk's are distinct, it is well known that Ln(Λ x) has exactly n zeros ln,n < ln-1,n < … < l2,n < l1,n on (0, 1).

First we prove the following global bound for the smallest zero

An important consequence is that if the associated Müntz space is non-dense in L2[0, 1], then

so the elements Ln(Λ x) have no zeros close to 0.

Furthermore, we determine the asymptotic behavior of the largest zeros; for k fixed,

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Research supported in part by NSF grants DMS-0400446 and DMS-0700427, and the Graduate Committee of the School of Mathematics at Georgia Institute of Technology.

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References
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[1] Almira, J. M., Möntz type theorems I. Surv. Approx. Theory 3 (2007), 152194.
[2] Borwein, P. and Erdélyi, T., Polynomials and polynomial inequalities. Graduate Texts in Mathematics, 161, Springer-Verlag, New York, 1995.
[3] Borwein, P., Erdélyi, T., and Zhang, J., Müntz systems and Müntz-Legendre polynomials. Trans. Amer. Math. Soc. 342 (1994), no. 2, 523542. http://dx.doi.org/10.2307/2154639
[4] Gurariy, V. I. and Lusky, W., Geometry of Müntz spaces and related questions. Lecture Notes in Mathematics, 1870, Springer-Verlag, Berlin, 2005.
[5] Lubinsky, D. S. and Saff, E. B., Zero distribution of Müntz extremal polynomials in Lp [0, 1] . Proc. Amer. Math. Soc. 135 (2007), no. 2, 427435. http://dx.doi.org/10.1090/S0002-9939-06-08694-1
[6] Olver, F.W. J., Asymptotics and special functions. A K Peters ,Wellesley, MA, 1997.
[7] Stef ánsson, Ú. F., Asymptotic behavior of Müntz orthogonal polynomials. Constr. Approx. 32 (2010), no. 2, 193220. http://dx.doi.org/10.1007/s00365-009-9059-x
[8] Stef ánsson, Ú. F., Endpoint asymptotics for Müntz-Legendre polynomials. Acta. Math. Hungar. 130 (2011), no. 4, 372381. http://dx.doi.org/10.1007/s10474-010-0013-y
[9] Stef ánsson, Ú. F., Zero spacing of Möntz orthogonal polynomials. Comput. Methods Funct. Theory. 11 (2011), no. 1, 4557
[10] Szegö, G., Orthogonal polynomials. American Mathematical Society, Colloquium Publications, 23, American Mathematical Society, Providence, RI, 1975.
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Canadian Mathematical Bulletin
  • ISSN: 0008-4395
  • EISSN: 1496-4287
  • URL: /core/journals/canadian-mathematical-bulletin
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