Higher-Order Szegő Expansions in Spectral Theory

• Higher-order Szegő expansions are advanced analytic identities that extend classical sum rules, capturing refined spectral and entropy properties in OPUC and operator theory.
• They employ polynomial filters, discrete Sobolev conditions, and canonical decompositions to quantify decay and regularity of Verblunsky coefficients in multi-singularity cases.
• Applications span spectral theory, random matrix models, and geometric analysis, providing trace formulas and asymptotic kernel expansions in both scalar and matrix contexts.

Higher-order Szegő expansions refer to sum rules, asymptotic expansions, and analytic identities that extend the classical Szegő theorem to capture finer spectral, functional, or geometric structure, typically involving weighted entropies, higher-order difference operators, matrix-valued or equivariant settings, or geometric kernels. They play a central role in spectral theory, orthogonal polynomials on the unit circle (OPUC), operator theory, and geometric analysis.

1. Classical Szegő Theorem and its Higher-Order Generalizations

The classical Szegő theorem links the relative entropy of a probability measure $\mu$ on the unit circle $\mathbb{T}$ to the $\ell^2$-norm of its Verblunsky coefficients $\{\alpha_n\}$: $$\int_0^{2\pi} \log w(\theta) \frac{d\theta}{2\pi} > -\infty \iff \sum_{n=0}^\infty |\alpha_n|^2 < \infty,$$ where $$d\mu(\theta) = w(\theta)\tfrac{d\theta}{2\pi} + d\mu_s$$ and $\mu_s$ is singular (Piao, 18 Jan 2026, Du, 2023).

Higher-order Szegő theorems replace the logarithmic integral with weighted forms involving trigonometric polynomials $Q(\theta)$ vanishing to order $m_k$ at singularities $\theta_k$. A prototypical higher-order sum rule is: $\mathbb{T}$0

The analytical structure for these theorems replaces the simple $\mathbb{T}$1-summability with decay and regularity conditions involving finite-band difference operators or more generally, discrete Sobolev regularity localized at prescribed points of the unit circle.

2. Algebraic and Analytic Framework for Higher-Order Expansions

Operator and Coefficient Expansions

Higher-order Szegő expansions rest on polynomial relations among the Verblunsky coefficients. Explicit combinatorial and recursive formulas have been established for Taylor coefficients of monic OPUC $\mathbb{T}$2 in terms of $\mathbb{T}$3 (Du, 2023): $\mathbb{T}$4 which, reorganized, yield closed-form, single-index expansions for moments of $\mathbb{T}$5 and support direct computational approaches to higher-order sum rules.

Canonical Algebraic Decompositions

In the multi-singularity case ($\mathbb{T}$6), Bézout identities in $\mathbb{T}$7 allow canonical splitting of every Verblunsky sequence $\mathbb{T}$8 so that each $\mathbb{T}$9 is filtered by $\ell^2$0, and precise interpolation and summability estimates (discrete Gagliardo–Nirenberg, Littlewood–Paley) yield optimal $\ell^2$1-decay results (Piao, 18 Jan 2026).

Spectral Representation and Operator-Theoretic Tools

The analysis of higher-order sum rules is closely connected to CMV matrix representations and traces of polynomial functions of CMV matrices. The expansion: $\ell^2$2 shows the entropy as a telescoping sum of local functionals of the coefficients and their higher-order differences (Piao, 18 Jan 2026).

3. Complete Characterization: The Simon-Lukic Conjecture

The Simon-Lukic conjecture for higher-order Szegő theorems, now affirmed in its most general form (Piao, 18 Jan 2026), states that for singularities $\ell^2$3 of order $\ell^2$4,

$\ell^2$5

if and only if $\ell^2$6 for $\ell^2$7 satisfying $\ell^2$8 and $\ell^2$9. This local-global dichotomy demonstrates that entropy is distributed among "resonant" filters centered at the singularities and that no further unconditional improvement is possible beyond the first-order case without additional $\{\alpha_n\}$0 control (Du, 2023).

A table summarizing the conditions at several levels:

Weight $\{\alpha_n\}$1	Necessary/Sufficient on $\{\alpha_n\}$2	Decomposition
$\{\alpha_n\}$3	$\{\alpha_n\}$4	None
$\{\alpha_n\}$5	$\{\alpha_n\}$6, $\{\alpha_n\}$7	None
$\{\alpha_n\}$8	$\{\alpha_n\}$9, $$\int_0^{2\pi} \log w(\theta) \frac{d\theta}{2\pi} > -\infty \iff \sum_{n=0}^\infty |\alpha_n|^2 < \infty,$$0	None
$$\int_0^{2\pi} \log w(\theta) \frac{d\theta}{2\pi} > -\infty \iff \sum_{n=0}^\infty |\alpha_n|^2 < \infty,$$1	Bézout-split $$\int_0^{2\pi} \log w(\theta) \frac{d\theta}{2\pi} > -\infty \iff \sum_{n=0}^\infty |\alpha_n|^2 < \infty,$$2: $$\int_0^{2\pi} \log w(\theta) \frac{d\theta}{2\pi} > -\infty \iff \sum_{n=0}^\infty |\alpha_n|^2 < \infty,$$3, $$\int_0^{2\pi} \log w(\theta) \frac{d\theta}{2\pi} > -\infty \iff \sum_{n=0}^\infty |\alpha_n|^2 < \infty,$$4	Canonical

4. Extensions: Matrix, Equivariant, and Geometric Szegő Expansions

Matrix-Valued Analogues

The matrix Szegő theorem generalizes scalar sum rules to measures with density $$\int_0^{2\pi} \log w(\theta) \frac{d\theta}{2\pi} > -\infty \iff \sum_{n=0}^\infty |\alpha_n|^2 < \infty,$$5 valued in positive semi-definite $$\int_0^{2\pi} \log w(\theta) \frac{d\theta}{2\pi} > -\infty \iff \sum_{n=0}^\infty |\alpha_n|^2 < \infty,$$6 matrices and matrix Verblunsky coefficients $$\int_0^{2\pi} \log w(\theta) \frac{d\theta}{2\pi} > -\infty \iff \sum_{n=0}^\infty |\alpha_n|^2 < \infty,$$7 satisfying $$\int_0^{2\pi} \log w(\theta) \frac{d\theta}{2\pi} > -\infty \iff \sum_{n=0}^\infty |\alpha_n|^2 < \infty,$$8 (Rouault, 2020). The higher-order sum rule becomes

$$\int_0^{2\pi} \log w(\theta) \frac{d\theta}{2\pi} > -\infty \iff \sum_{n=0}^\infty |\alpha_n|^2 < \infty,$$9

with $$d\mu(\theta) = w(\theta)\tfrac{d\theta}{2\pi} + d\mu_s$$0 an explicit trace correction term. In the case of commuting or diagonalizable $$d\mu(\theta) = w(\theta)\tfrac{d\theta}{2\pi} + d\mu_s$$1, the theorem reduces to scalar components.

Equivariant and Geometric Expansions

On CR manifolds with group actions, higher-order Szegő expansions describe the asymptotics of equivariant Szegő kernels with explicit coefficients. For a compact strongly pseudoconvex CR manifold $$d\mu(\theta) = w(\theta)\tfrac{d\theta}{2\pi} + d\mu_s$$2 with $$d\mu(\theta) = w(\theta)\tfrac{d\theta}{2\pi} + d\mu_s$$3 and compact Lie group $$d\mu(\theta) = w(\theta)\tfrac{d\theta}{2\pi} + d\mu_s$$4 action, the $$d\mu(\theta) = w(\theta)\tfrac{d\theta}{2\pi} + d\mu_s$$5-th irreducible component of the $$d\mu(\theta) = w(\theta)\tfrac{d\theta}{2\pi} + d\mu_s$$6-th Fourier Szegő projection admits a complete expansion (Hsiao et al., 2020): $$d\mu(\theta) = w(\theta)\tfrac{d\theta}{2\pi} + d\mu_s$$7 with closed-form formulas for $$d\mu(\theta) = w(\theta)\tfrac{d\theta}{2\pi} + d\mu_s$$8, $$d\mu(\theta) = w(\theta)\tfrac{d\theta}{2\pi} + d\mu_s$$9 in terms of Tanaka–Webster scalar curvature, moment map, Laplacians, group characters, and orbit volumes. Higher-order coefficients are explicit universal polynomials in the local geometry and group invariants.

Real Ball and Kernel Expansions

Szegő-type expansions also arise for harmonic and holomorphic reproducing kernels on domains such as the ball $\mu_s$0. The Szegő kernel for $\mu_s$1-harmonic functions admits a triple hypergeometric series expansion, finite-sum Gauss representation, and, for collinear arguments, an Appell $\mu_s$2 reduction (Moravík, 13 Oct 2025). These formulas are adapted to precise boundary behavior and are central for radial Toeplitz operators and weighted Bergman kernel asymptotics.

5. Remainder Estimates, Sharpness, and Failure of Unconditional “Gems”

Analysis of boundary terms and error bounds shows that the positive-definite structure of the filtered $\mu_s$3 (or $\mu_s$4) summability is optimally sharp (Du, 2023, Piao, 18 Jan 2026). Remainder terms in truncated higher-order expansions scale with high powers of $\mu_s$5 or filtered combinations and are absolutely summable under the canonical hypotheses. Any relaxation (for instance, seeking a sum rule involving only the filtered $\mu_s$6 term) generates counterexamples with diverging entropy, affirming that further unconditional spectral gems are not possible beyond the first-order case.

6. Applications and Consequences in Operator Theory and Random Matrix Models

Higher-order Szegő expansions play a vital role in:

• Spectral theory of CMV and block Toeplitz operators: The expansions yield trace-class, Hilbert–Schmidt, and Schatten–von Neumann criteria for perturbations, directly translating into scattering and spectral results for block Jacobi and CMV matrices (Rouault, 2020).
• Random matrix theory: Higher-order sum rules control the large deviation rate function and partition function asymptotics in models such as Gross–Witten.
• Geometric analysis: Asymptotic Szegő kernel expansions on CR manifolds quantify curvature effects, symmetries, and quantization corrections (Hsiao et al., 2020, Paoletti, 2016).
• Multichannel and signal processing: Matrix Szegő expansions underpin scattering theory and information-theoretic functionals in multi-input/multi-output systems (Rouault, 2020).

The unified analytic, algebraic, and geometric framework underlying higher-order Szegő expansions synthesizes operator-theoretic, combinatorial, and harmonic-analytic tools and represents a foundational principle in the contemporary spectral theory of unitary operators and geometric quantization.