A local discrepancy theorem

Abstract

Let σ be a signed measure on [−1,1], with total mass 0, and [ a,b] ⊆ [−1,1]. We estimate |σ([ a,b])| in terms of the logarithmic potential of σ in a complex neighborhood of [ a,b]. Applications include a detailed description of the distribution of zeros of a class of orthogonal polynomials, including Pollaczek polynomials.