A Complete Proof of the Simon--Lukic Conjecture for Higher-Order Szeg\H{o} Theorems

TL;DR

This paper proves the Simon-Lukic conjecture for higher-order Szeg ext{"o} theorems, linking entropy conditions with Verblunsky coefficients and spectral measure decompositions for orthogonal polynomials on the unit circle.

Contribution

It provides the first complete proof of the conjecture, introducing new algebraic decomposition techniques and synthesizing advanced mathematical tools.

Findings

Established the equivalence between entropy and decomposition conditions.

Proved the conjecture for measures with singular points and multiplicities.

Demonstrated the local-global behavior in spectral theory.

Abstract

This paper provides a complete proof of Simon-Lukic conjecture for orthogonal polynomials on the unit circle. For a probability measure $d\mu = w(\theta) \frac{d\theta}{2\pi} + d\mu_s$ with Verblunsky coefficients $\alpha=\{\alpha_n\}_{n=0}^\infty$, distinct singular points $(\theta_k)_{k=1}^{\ell}$, and multiplicities $(m_k)_{k=1}^{\ell}$, we establish the equivalence between the entropy condition \[ \int_0^{2\pi} \prod_{k=1}^{\ell} [1 - \cos(\theta - \theta_k)]^{m_k} \log w(\theta) \frac{d\theta}{2\pi} > -\infty \] and the decomposition condition \[ \exists \beta^{(1)}, \ldots, \beta^{(\ell)} : \alpha = \sum_{k=1}^\ell \beta^{(k)} \,\, \text{with} \,\, (S - e^{-i\theta_k})^{m_k} \beta^{(k)} \in \ell^2, \,\, \beta^{(k)} \in \ell^{2m_k + 2}. \] The proof synthesizes unitary transformations, discrete Sobolev-type inequalities, higher-order Szeg\H{o} expansions, and a novel algebraic decomposition technique. Our resolution affirms that spectral theory is fundamentally local-global behavior emerges from the superposition of local resonances, each governed by its intrinsic scale.