Single-Point Higher-Order Szeg\H{o} Sum Rules in OPUC: Necessity for $m=1,2,3$

Abstract

We give a direct algebraic proof of the necessity direction in the single-point higher-order Szeg\H{o} sum rules on the unit circle for $m=1,2,3$. More precisely, for $H_m(e^{i\theta})=(1-\cos\theta)^m$, we show that $\int_0^{2\pi}H_m(e^{i\theta})\log w(\theta)\frac{d\theta}{2\pi}>-\infty$ implies $(S-1)^m\alpha\in\ell^2,\qquad \alpha\in\ell^{2m+2}.$ The proof is carried out within Yan's algebraic model for higher-order sum rules. The main point is to obtain coercive lower bounds for the nonlogarithmic part of the truncated sum rule: the quadratic component yields the principal finite-difference energy, while the higher-order correction terms are controlled by telescoping cancellations and relative bounds. The logarithmic remainder then gives the required $\ell^{2m+2}$-summability. The purpose is to isolate explicit low-order necessity arguments within the algebraic framework.