RATIONAL APPROXIMATION WITH VARYING WEIGHTS .1.

We investigate two problems concerning uniform approximation by weight ed rationals {w(n)r(n)}(infinity)(n=1), where r(n) = p(n)/q(n) is a ra tional function of order n. Namely, for w(x) := e(x) we prove that uni form convergence to 1 of w(n)r(n) is not possible on any interval [0, alpha] with a > 2 pi. For w(x) := x(theta), theta > 1, We show that un iform convergence to 1 of w(n)r(n) is not possible on any interval [b, 1] with b < tan(4)(pi(theta - 1)/4 theta). (The latter result can be interpreted as a rational analogue of results concerning ''incomplete polynomials.'') More generally, for alpha greater than or equal to 0, beta greater than or equal to 0, alpha + beta > 0, we investigate for w(x) = e(x) and w(x) = x(theta), the possibility of approximation by { w(n)p(n)/q(n)}(infinity)(n=1), where deg p(n) less than or equal to 5 alpha n, deg q(n) beta n. The analysis utilizes potential theoretic me thods. These are essentially sharp results though this will not be est ablished in this paper.