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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. In a separate real signed-measure recipient, a negative interval gives whole negative polynomial spaces of order (log N)^(1−α) for 0 ≤ α < 1, and log log 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

The full-strip core uses the stated finite-order divisor and combined Carleman moment assumptions. The inertia hierarchy concerns even finite signed measures with a supplied negative interval; the matching orders are proved in that finite-measure class. The all-depth application concerns the specified analytic kernel family and its actual supremum weight.

Disclosures

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

Reproduction

The package contains the complete nineteen-page manuscript and all nine editable TeX modules. Build with Tectonic 0.17.0 using tectonic main.tex. The results here have analytic proofs. The inherited all-depth nonreal-zero input cites public edition 2 of All-depth root-Laguerre positivity with nonreal Fourier zeros, including both whole-line cumulant certificates. README.md explains the dependencies.

References and related resources

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