A Complete Proof of the Simon--Lukic Conjecture for Higher-Order Szegő Theorems
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866908784855613440 |
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| author | Piao, Daxiong |
| author_facet | Piao, Daxiong |
| contents | This paper provides a complete proof of Simon-Lukic conjecture for orthogonal polynomials on the unit circle. For a probability measure $dμ= w(θ) \frac{dθ}{2π} + dμ_s$ with Verblunsky coefficients $α=\{α_n\}_{n=0}^\infty$, distinct singular points $(θ_k)_{k=1}^{\ell}$, and multiplicities $(m_k)_{k=1}^{\ell}$, we establish the equivalence between the entropy condition \[ \int_0^{2π} \prod_{k=1}^{\ell} [1 - \cos(θ- θ_k)]^{m_k} \log w(θ) \frac{dθ}{2π} > -\infty \] 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2601_12332 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Complete Proof of the Simon--Lukic Conjecture for Higher-Order Szegő Theorems Piao, Daxiong Spectral Theory 42C05, 47B36, 60F10, 41A60, 33C47 This paper provides a complete proof of Simon-Lukic conjecture for orthogonal polynomials on the unit circle. For a probability measure $dμ= w(θ) \frac{dθ}{2π} + dμ_s$ with Verblunsky coefficients $α=\{α_n\}_{n=0}^\infty$, distinct singular points $(θ_k)_{k=1}^{\ell}$, and multiplicities $(m_k)_{k=1}^{\ell}$, we establish the equivalence between the entropy condition \[ \int_0^{2π} \prod_{k=1}^{\ell} [1 - \cos(θ- θ_k)]^{m_k} \log w(θ) \frac{dθ}{2π} > -\infty \] 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. |
| title | A Complete Proof of the Simon--Lukic Conjecture for Higher-Order Szegő Theorems |
| topic | Spectral Theory 42C05, 47B36, 60F10, 41A60, 33C47 |
| url | https://arxiv.org/abs/2601.12332 |