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Müntz systems and orthogonal Müntz-Legendre polynomials. (English) Zbl 0799.41015
Summary: The Müntz-Legendre polynomials arise by orthogonalizing the Müntz system $$\{x^{\alpha_ 0}, x^{\alpha_ 1}, \dots\}$$ 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 tends 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.

MSC:
 41A17 Inequalities in approximation (Bernstein, Jackson, Nikol’skiĭ-type inequalities) 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis 30C15 Zeros of polynomials, rational functions, and other analytic functions of one complex variable (e.g., zeros of functions with bounded Dirichlet integral) 39A10 Additive difference equations
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