Admissible kernel lifts with positive endpoint forms and nonreal Fourier zeros
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Abstract
We give a finite-atomic construction of admissible Fourier kernels with 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 produces kernels satisfying these shape conditions whose physical endpoint moment matrices are positive definite at every finite order, while their Fourier transforms have nonreal zeros. 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. 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 finite-atomic global curvature and centered-cumulant lifting theorem, root-coordinate and second-level shape inequalities, and a fully admissible order-one counterexample to propagation from all endpoint moment orders to interior physical positivity. For each prescribed strip a different kernel is constructed.
Disclosures
Research and manuscript preparation used AI assistance. Classical Bessel, Laguerre, Turán and Jensen results and earlier author publications are credited in the manuscript.
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