A Complete Proof of the Simon--Lukic Conjecture for Higher-Order Szegő Theorems

Abstract: This paper provides a complete proof of Simon-Lukic conjecture for orthogonal polynomials on the unit circle. For a probability measure $dμ= w(θ) \frac{dθ}{2π} + dμs$ with Verblunsky coefficients $α={α_n}{n=0}^\infty$, distinct singular points $(θk){k=1}^{\ell}$, and multiplicities $(m_k){k=1}^{\ell}$, we establish the equivalence between the entropy condition [ \int_0^{2π} \prod{k=1}^{\ell} [1 - \cos(θ- θk)]^{m_k} \log w(θ) \frac{dθ}{2π} > -\infty ] and the decomposition condition [ \exists β^{(1)}, \ldots, β^{(\ell)} : α= \sum{k=1}^\ell β^{(k)} \,\, \text{with} \,\, (S - e^{-iθ_k})^{m_k} β^{(k)} \in \ell^2, \,\, β^{(k)} \in \ell^{2m_k + 2}. ] The proof synthesizes unitary transformations, discrete Sobolev-type inequalities, higher-order Szegő 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.