MUNTZ SYSTEMS AND ORTHOGONAL MUNTZ-LEGENDRE POLYNOMIALS

The Muntz-Legendre polynomials arise by orthogonalizing the Muntz syst em {x(lambda0), x(lambda1), ...} with respect to Lebesgue measure on [ 0, 1]. In this paper, differential and integral recurrence formulae fo r the Muntz-Legendre polynomials are obtained. Interlacing and lexicog raphical properties of their zeros are studied, and the smallest and l argest zeros are universally estimated via the zeros of Laguerre polyn omials. The uniform convergence of the Christoffel functions is proved equivalent to the nondenseness of the Muntz space on [0, 1], which im plies that in this case the orthogonal Muntz-Legendre polynomials tend to 0 uniformly on closed subintervals of [0, 1). Some inequalities fo r Muntz polynomials are also investigated, most notably, a sharp L2 Ma rkov inequality is proved.