Higher-Order Szego Theorems With Two Singular Points

2004/09/23 by Barry Simon, Simon, Barry, Andrej Zlatos +1 · 2 citations
Mathematics · Physics and Astronomy · #Classical Analysis and ODEs (math.CA) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #math-ph #math.CA #math.MP

paper · pdf · doi:10.48550/arxiv.math-ph/0409065

arxiv created 2004/09/23 · arxiv updated 2009/12/01

Abstract

We consider probability measures, dμ=w(θ) \fdθ2π +dμ_\s, on the unit circle, ∂\bbD, with Verblunsky coefficients, \α_j\_j=0^∞. We prove for θ₁≠θ₂ in [0,2π) and (δβ)_j=β_j+1 that ∫ [1-cos(θ-θ₁)][1-cos(θ-θ₂)] log w(θ) \fdθ2π >-∞ if and only if ∑_j=0^∞ |\(δ-e^-iθ₂) (δ-e^-iθ₁) α\_j|² +\absα_j⁴ <∞ We also prove that ∫ (1-cosθ)² log w(θ) \fdθ2π >-∞ if and only if ∑_j=0^∞ \absα_j+2-2α_j+1 +α_j² + \absα_j⁶ <∞

Cited by

• A Mixed-Resonance Counterexample to the Lukic Conjecture
• Sum rules via large deviations: polynomial potentials and multi-cut regime on the unit circle

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