Bounds on Orthonormal Polynomials for Restricted Measures

Suppose that ν is a given positive measure on [ -1,1] , and that μ is another measure on the real line, whose restriction to ( -1,1) is ν . We show that one can bound the orthonormal polynomials p_n( μ ,y) for μ and y∈ℝ , by the supremum of | S_J( y) p_n-J( S_J^2ν ,y) | , where the sup is taken over all 0≤ J≤ n and all monic polynomials S_J of degree J with zeros in an appropriate set.