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Polynomial cores and inertia under Carleman moment bounds

Scott Schubr

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Abstract

We study ordinary polynomial approximation in a joint norm containing a holomorphic product derivative on a full horizontal strip and a real-axis observation. A Carleman condition on the combined horizontal moments gives a polynomial core for finite-order entire divisors, with every interior zero multiplicity retained and with no boundary-zero separation. A finite moment version gives an explicit terminal-clock estimate for the original complex-coordinate graph. For factorial--logarithmic moment budgets we prove a universal core guarantee through exponent one and give finite-order, real-zero counterexamples beyond it, using their own physical observation weights. For a separate real signed-measure quadratic form, a negative interval gives whole negative polynomial spaces of order (log N)^(1-α) for 0≤α<1, and loglog N at α=1, in each parity. Slowly varying Jacobi models give matching orders in this finite-measure class. Finally, a double-exponential spatial kernel has the required square-root-logarithmic Fourier moments for its actual physical weight. Its joint core at every width coexists with growing negative physical polynomial inertia at later widths, despite its positive all-depth root hierarchy. Classical quasianalyticity, Dirichlet cyclicity, scalar approximation and orthogonal-polynomial results are explicitly credited.

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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
No additional permission granted

Scope

Finite-order entire divisors with the stated Carleman horizontal moments have ordinary-polynomial same-width joint zero-jet cores. Factorial-logarithmic budgets give a sharp universal exponent-one threshold, quantitative negative polynomial spaces under a supplied negative band and matching finite-measure Jacobi orders.

Disclosures

AI tools assisted mathematical development, source preparation and internal checking. The author is responsible for the statements and proofs.

Reproduction

The 19-page manuscript and editable sources are supplied in this 13-member package. README.md gives Tectonic 0.17.0 build instructions for the nine TeX files. The complete analytic proofs and explicit constants are supplied; no computational programs or external datasets are required. The nonreal-zero application cites its inherited whole-line cumulant certificates in the specified public all-depth predecessor edition 2.

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.