LOCAL ASYMPTOTIC-DISTRIBUTION OF ZEROS OF ORTHOGONAL POLYNOMIALS

Converse results, which state a relation (inequality) for measures fro m that on their logarithmic potentials, are applied to local density o f zeros of orthogonal polynomials when the measure of orthogonality is a general one with compact support. It will be shown that if the meas ure is sufficiently thick on a part of its support, then on that part the density of the zeros will be at least as large as the equilibrium measure of the support. A corresponding upper estimate on the distribu tion of the zeros will also be proved. All of our estimates are sharp, and they localize several well-known results.