Luminary Research Archive
Luminary Research ArchiveIndependent scholarship
Luminary Portal
PublicEdition 3

Polynomial cores and inertia under Carleman moment bounds

Scott Schubr ORCID

Original PDF · First-page preview

Loading original first page…

Abstract

We establish a universal threshold for polynomial approximation in a joint graph norm containing a holomorphic product derivative on a full horizontal strip, an anchor value and an original real-axis observation. For a positive log-convex moment envelope m_n, the divergence of the sum of m_(n−1)/m_n guarantees a polynomial core under finite-order divisor hypotheses. When this sum converges, real-zero entire functions of order one satisfy the same envelope for their full graph and their own physical observation weights but fail to have a polynomial core. A separate signed countermodel has a negative interval while every nonzero polynomial square has positive integral. The core theorem also yields same-width synthesis for the actual Riemann xi-function at every positive strip width, retaining every interior zero multiplicity and permitting boundary zeros. Uniform physical-potential and normalized substrip-energy convergence follow. Quantitative finite-moment graph estimates and signed-measure inertia bounds give whole negative polynomial spaces on logarithmic and iterated-logarithmic scales, with matching finite-measure Jacobi models. A double-exponential spatial kernel realizes the joint-core and negative-inertia phenomena with its actual physical weight. Classical quasianalyticity, cyclicity, Fourier support and orthogonal-polynomial results supply the underlying tools.

Read the full paper in the original PDF.

Edition files

Available editions

  1. Edition 3 · 2026-10-30 · public
  2. Edition 2 · 2026-10-27 · public
  3. Edition 1 · 2026-10-26 · public

Research details

Subjects
Mathematics · Complex Analysis · Functional Analysis · Spectral Theory
Keywords
Carleman moment bounds, polynomial approximation, holomorphic strip, Dirichlet space, signed measures, negative inertia, Jacobi matrices
Manuscript date
2026-10-26
Reader reuse
none

Scope

Ordinary-polynomial same-width joint zero-jet cores for finite-order entire divisors obey a sharp universal log-convex moment-envelope criterion. The convergent-envelope construction retains the divisor’s own physical weight, positive physical potentials, real zeros and full graph moments. The actual Riemann xi-function admits compatible-target synthesis at every positive strip width. Finite moment clocks, whole negative polynomial spans and matching finite-measure models give quantitative applications.

Disclosures

Author-posted preprint. OpenAI ChatGPT and Codex assisted mathematical derivations, proof development and checking, literature discovery and comparison, computational verification, source preparation, and manuscript drafting and revision. Scott Schubr is the sole author and takes responsibility for the entire content. The author owns Luminary Oddities and the Luminary Research Archive and declares no other competing interests. This research received no external funding.

Reproduction

The 23-page manuscript and complete editable modular sources are supplied in the 15-member source package with build instructions and a complete member manifest. All general-envelope and same-width xi assertions have analytic proofs in the manuscript. The kernel application cites its inherited whole-line rational cumulant certificates in the specified predecessor; no new empirical dataset or computational program is required.

References and related resources

  1. M. de Jeu, Determinate multidimensional measures, the extended Carleman theorem and quasi-analytic weights, Ann. Probab. 31 (2003), 1205–1227.
  2. M. de Jeu, Subspaces with equal closure, Constr. Approx. 20 (2004), 93–157.
  3. F. Nazarov, M. Sodin and A. Volberg, Lower bounds for quasianalytic functions, I. How to control smooth functions?, Math. Scand. 95 (2004), 59–79.
  4. F. Koehler and B. Wu, Constructive Approximation under Carleman’s Condition, with Applications to Smoothed Analysis (2025).
  5. S. Richter and C. Sundberg, Multipliers and invariant subspaces in the Dirichlet space, J. Operator Theory 28 (1992), 167–186.
  6. O. El-Fallah, K. Kellay and T. Ransford, On the Brown–Shields conjecture for cyclicity in the Dirichlet space (2008).
  7. A. Aleman, M. Hartz, J. E. McCarthy and S. Richter, The Smirnov class for spaces with the complete Pick property (2017).
  8. M. T. Jury and R. T. W. Martin, Unbounded multipliers of complete Pick spaces (2021).
  9. A. E. Ingham, A note on Fourier transforms, J. London Math. Soc. s1-9 (1934), 29–32.
  10. S. Bagchi, P. Ganguly, J. Sarkar and S. Thangavelu, An analogue of Ingham’s theorem on the Heisenberg group (2022).
  11. G. Świderski and B. Trojan, Asymptotic behaviour of Christoffel–Darboux kernel via three-term recurrence relation I, Constr. Approx. 54 (2021), 49–116; theorem numbering from arXiv v2.
  12. S. Schubr, Polynomial Recovery and Masking under Weak Exponential Tails, edition 1 (2026).
  13. S. Schubr, Polynomial inertia of signed measures with negative intervals, edition 1 (2026).
  14. S. Schubr, All-depth root-Laguerre positivity with nonreal Fourier zeros, edition 2 (2026), including both whole-line cumulant enclosures.
  15. S. Schubr, Polynomial synthesis and zero ideals for the Riemann xi-function, Luminary Research Archive, preprint, edition 2 (2026).