A counterexample and corrected translation-density criterion for correlation kernels
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Abstract
A displayed correlation-kernel characterization of the Laguerre–Pólya class in Dimitrov and Xu, Wronskians of Fourier and Laplace transforms, fails even when the Fourier transform has only simple real zeros. We construct an even nonnegative compactly supported smooth probability density whose order-one transform has a pure paired Hadamard product and only simple real zeros, but whose stated correlation kernel has non-dense translates in L^1(ℝ) at the explicit width 1/1000. The sign change is proved by an analytic bound for the entire infinite product, without numerical evaluation of that product. The underlying distinction is between an outer hyperbolic factor, which receives the real part of an analytic Wronskian, and an inner hyperbolic factor, which receives the mixed Jensen curvature. We prove the corrected density equivalence, including its width-zero simplicity endpoint, under explicit moment and paired-product assumptions. The Jensen and Wiener principles are classical.
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Research details
- Subjects
- Mathematics · Classical and Real Analysis · Complex Analysis
- Keywords
- Fourier transform, translation density, Laguerre-Polya class, Jensen curvature, correlation kernel
- Manuscript date
- 2026-10-25
- Reader reuse
- No additional permission granted
Scope
An explicit smooth simple-real-zero counterexample to the displayed correlation-kernel translation-density claim, and a corrected equivalence under stated paired-product and moment assumptions.
Disclosures
AI tools assisted with mathematical development, literature comparison, proof checking and manuscript preparation. The author is responsible for the statements and proofs. This preprint does not claim external peer review or formal proof verification.
Reproduction
7-page manuscript and 7 source-package members. Compile main.tex with the included build.py and Tectonic. Standard-library Python runs python -B verify_counterexample.py, also under -O, to check the exact Wronskian derivative and rational sign bounds.