A correction to a correlation-kernel density criterion
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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 L1(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
A source-specific correction to the literal correlation-kernel formula and universal necessity implication, with a smooth globally simple-zero counterexample and repaired criterion under explicit hypotheses. This is an independent correction note.
Disclosures
AI tools assisted with mathematical development, literature comparison, proof checking and manuscript preparation. The author is responsible for the statements and proofs.
Reproduction
The complete package contains the manuscript, editable LaTeX, a portable build and an exact-arithmetic supplement. The analytic infinite-product and density proofs are accompanied by finite derivative and rational-margin controls.