Higher-Order Szego Theorems With Two Singular Points

Abstract

We consider probability measures, $dμ=w(θ) \f{dθ}{2π} +dμ_\s$, on the unit circle, $\partial\bbD$, with Verblunsky coefficients, $\{α_j\}_{j=0}^\infty$. We prove for $θ_1\neqθ_2$ in $[0,2π)$ and $(δβ)_j=β_{j+1}$ that \[ \int [1-\cos(θ-θ_1)][1-\cos(θ-θ_2)] \log w(θ) \f{dθ}{2π} >-\infty \] if and only if \[ \sum_{j=0}^\infty \bigl|\bigl\{(δ-e^{-iθ_2}) (δ-e^{-iθ_1}) α\bigr\}_j\bigr|^2 +\abs{α_j}^4 <\infty \] We also prove that \[ \int (1-\cosθ)^2 \log w(θ) \f{dθ}{2π} >-\infty \] if and only if \[ \sum_{j=0}^\infty \abs{α_{j+2}-2α_{j+1} +α_j}^2 + \abs{α_j}^6 <\infty \]