Müntz systems and orthogonal Müntz-Legendre polynomials

Abstract

The Müntz-Legendre polynomials arise by orthogonalizing the Müntz system { x λ 0 , x λ 1 , … } \{ {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 {L^2} Markov inequality is proved.