UNIFORM-CONVERGENCE OF POLYNOMIALS ASSOCIATED WITH VARYING JACOBI WEIGHTS

In this paper we determine the functions on [-1, 1] that are uniform limits of weighted polynomials of the form (1 - x)-alpha-n (1 + x)-beta-n p(n)(x), where deg p(n) less-than-or-equal-to n, lim(n) --> infinity alpha-n/n = theta-1 greater-than-or-equal-to 0 and lim(n) --> infinity beta-n/n = theta-2 greater-than-or-equal-to 0. Estimates for the rate of convergence are also obtained. Our results confirm a conjecture of Saff for w(x) = (1 - x)-theta-1 (1 + x)-theta-2, when theta-1 > 0, theta-2 > 0, and extend previous results of G.G. Lorentz and M. v. Golitschek, and Staff and Varga for incomplete polynomials.