Admissible kernel lifts with positive endpoint forms and nonreal Fourier zeros
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Abstract
We construct admissible Fourier kernels whose physical endpoint moment matrices are positive definite at every finite order while their Fourier transforms have nonreal zeros. The construction uses prescribed Fourier multipliers of positive finite atomic measures. A sufficiently broad hyperbolic-cosh base makes the translated mixture uniformly strictly log-concave while retaining an order-one entire Fourier transform. A global centered-moment and cumulant estimate also preserves strict concavity of G(u)=log φ(√(u)) and of log(S'(u)^2-S(u)S''(u)), where S(u)=φ(√(u)). The required scales are explicit and conservative. An exact four-atom multiplier realizes this separation under all these shape conditions. All zeros can be confined to any prescribed horizontal strip; an interior physical parameter still has negative polynomial squares in both parities. Classical Turán consequences are stated with their coefficient normalization and do not imply real-zero closure. The construction strengthens a generic nonimplication within the full admissible class.
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Research details
- Subjects
- Mathematics · Complex Analysis
- Keywords
- admissible Fourier kernels, Laguerre inequalities, log-concavity, entire functions, moment matrices, nonreal zeros
- Manuscript date
- 2026-10-24
- Reader reuse
- No additional permission granted
Scope
Explicit admissible Fourier-kernel lifts with all finite endpoint moment matrices positive, additional root-coordinate shape conditions and nonreal zeros confined to a prescribed horizontal strip.
Disclosures
Research and manuscript preparation used AI assistance. This author preprint has not undergone external human refereeing. Classical Bessel, Laguerre, Turán and Jensen results and earlier author publications are credited in the manuscript.
Reproduction
12-page manuscript and 12 source-package members. Compile main.tex. Standard-library python verify_lifting_constants.py, also under -O, checks the explicit conservative lifting and shape-inequality constants accompanying the analytic construction.