Functions of the Second Kind for Freud Weights and Series Expansions of Hilbert Transforms

Abstract

Let W ≔ e − Q , where Q: R → R is even, continuous, and of smooth polynomial growth at infinity. Then we call W 2 = e −2 Q a Freud weight, the most typical examples being W 2 β( x) ≔ exp(−| x| β), β > 1. Corresponding to the weight W 2, we can form the sequence of orthonormal polynomials { p j ( W 2, x)} ∞ j=0 . The functions of the second kind are q j ( W 2, x) ≔ H[ p j W 2]( x) j ≥ 0, where H denotes the Hilbert transform; that is, for g ∈ L, ( R ), H[ g]( x) ≔ P.V. ∫ ∞ −∞ g( t)/ t − x dt. Here P.V. denotes principal value. For a large class of Freud weights, we obtain bounds on { q j } ∞ j=0 in the L ∞, and L p norms. We also estimate the generalized function of the second kind q j ( W 2, v, x) ≔ H[ p j W v]( x), for a fixed function v. We then apply these estimates to investigate the convergence of series of the second kind, which form the basis of Henrici′s method of approximating Cauchy principal value integrals.