Single-Point Higher-Order Szegő 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 Szegő 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 $(S-1{)}^m\alpha \in {\ell }^2,\alpha \in {\ell }^{2m+2}.$ 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.