Mntz systems and orthogonal Mntz-Legendre polynomials
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The Mntz-Legendre polynomials arise by orthogonalizing the Mntz system with respect to Lebesgue measure on [0, 1]. In this paper, differential and integral recurrence formulae for the Mntz-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 Mntz space on [0, 1], which implies that in this case the orthogonal Mntz-Legendre polynomials tend to 0 uniformly on closed subintervals of [0, 1). Some inequalities for Mntz polynomials are also investigated, most notably, a sharp Markov inequality is proved.