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Published May 2005 | Submitted
Journal Article Open

Higher-order Szegő theorems with two singular points

Abstract

We consider probability measures, dμ = w(θ)^(dθ)_(2π) + dμ_s, on the unit circle, ∂D, with Verblunsky coefficients, {αj}_(j=0)^∞. We prove for θ_1 ≠ θ_2 in [0,2π) that ∫[1-cos(θ-θ_1)][1-cos(θ-θ_2)]log w(θ)^(dθ)_(2π > -∞if and only if ∑_(j=0)^∞ │{(δ-e^(-iθ2))(δ-e^(-iθ1))α}_j^2 +|α_j|^4 < ∞,where δ is the left shift operator (δβ)_j = β_(j+1). We also prove that ∫(1-cosθ)^2 log w (θ)^(dθ)_(2π) > - ∞ if and only if ∑_(j=0)^∞|α_(j+2) - 2α_(j+1) + α_j|^2 + |αj|^ 6 <∞.

Additional Information

© 2005 Elsevier Inc. Received 16 September 2004, Accepted 9 February 2005, Available online 7 April 2005. Communicated by Leonid Golinskii We thank S. Denisovand S. Kupin for telling us of their joint work [3].

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August 22, 2023
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October 25, 2023