Single-Point Higher-Order Szegő Sum Rules in OPUC: Necessity for $m=1,2,3$
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866914506470326272 |
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| author | Piao, Daxiong |
| author_facet | Piao, Daxiong |
| contents | 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 $H_m(e^{iθ})=(1-\cosθ)^m$, we show that $\int_0^{2π}H_m(e^{iθ})\log w(θ)\frac{dθ}{2π}>-\infty$ implies $(S-1)^mα\in\ell^2,\qquad α\in\ell^{2m+2}.$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2604_23032 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Single-Point Higher-Order Szegő Sum Rules in OPUC: Necessity for $m=1,2,3$ Piao, Daxiong Classical Analysis and ODEs Spectral Theory 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 $H_m(e^{iθ})=(1-\cosθ)^m$, we show that $\int_0^{2π}H_m(e^{iθ})\log w(θ)\frac{dθ}{2π}>-\infty$ implies $(S-1)^mα\in\ell^2,\qquad α\in\ell^{2m+2}.$ 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. |
| title | Single-Point Higher-Order Szegő Sum Rules in OPUC: Necessity for $m=1,2,3$ |
| topic | Classical Analysis and ODEs Spectral Theory |
| url | https://arxiv.org/abs/2604.23032 |