The generalized functions of the second kind for general exponential weights

Abstract

Let I be a finite or infinite interval containing 0, and let W : I → (0, ∞). Assume that W 2 is a weight, so that we may define orthonormal polynomials corresponding to W 2. The generalized functions of the second kind are , where H denotes the Hilbert transform, and v the bounded function on I. We investigate boundedness of the sum of qj (W 2, v, x)2 for the general class of exponential weights