For single-critical-point weights H_m = (1-cos θ)^m, the condition ∫ (1-cos θ)^m log w dθ/2π > -∞ implies Δ^m α ∈ ℓ² and α ∈ ℓ^{2m+2}.
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Necessary Conditions for Single-Critical-Point Higher-Order Szeg\H{o} Sum Rules in OPUC
For single-critical-point weights H_m = (1-cos θ)^m, the condition ∫ (1-cos θ)^m log w dθ/2π > -∞ implies Δ^m α ∈ ℓ² and α ∈ ℓ^{2m+2}.