Approximation by Polynomials with Restricted Zeros

This paper discusses convergence properties of polynomials whose zeros lie on the real axis or in the upper half-plane. A result of Levin shows that uniform convergence of such polynomials to a non-zero limit on a complex sequence converging not too East to a limit in the lower half-plane implies locally uniform convergence in C. We give a relatively simple proof of this result and present several extensions and examples which show that the criterion in Levin′s theorem is almost sharp.