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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Müntz systems and orthogonal Müntz-Legendre polynomials

Authors: Peter Borwein, Tamás Erdélyi and John Zhang
Journal: Trans. Amer. Math. Soc. 342 (1994), 523-542
MSC: Primary 42C05; Secondary 39A10, 41A17
MathSciNet review: 1227091
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Abstract: The Müntz-Legendre polynomials arise by orthogonalizing the Müntz system $ \{ {x^{{\lambda _0}}},{x^{{\lambda _1}}}, \ldots \} $ with respect to Lebesgue measure on [0, 1]. In this paper, differential and integral recurrence formulae for the Müntz-Legendre polynomials are obtained. Interlacing and lexicographical properties of their zeros are studied, and the smallest and largest zeros are universally estimated via the zeros of Laguerre polynomials. The uniform convergence of the Christoffel functions is proved equivalent to the nondenseness of the Müntz space on [0, 1], which implies that in this case the orthogonal Müntz-Legendre polynomials tend to 0 uniformly on closed subintervals of [0, 1). Some inequalities for Müntz polynomials are also investigated, most notably, a sharp $ {L^2}$ Markov inequality is proved.

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Keywords: Exponential polynomials, Müntz systems, Müntz-Szász Theorem, Müntz-Legendre polynomials, recurrence formulae, orthogonal polynomials, Christoffel functions, Markov-type inequalities, Bernstein-type inequalities, Nikolskii-type inequalities, interlacing properties of zeros, lexicographic properties of zeros
Article copyright: © Copyright 1994 American Mathematical Society