A strictly PF2 Green kernel with nonreal polynomial roots

Abstract

We give an explicit counterexample to the polynomial converse posed by Albert and Levandosky (arXiv:2603.23876v1, Remark 3.7). For p(z) = (z+1)(z+4)((z+8)²+36)/400, the Fourier symbol p(ξ²) is positive and elliptic of degree eight, although p has nonreal zeros. Its inverse Fourier kernel is a positive, even C⁶ probability density K satisfying (log K)″ < 0 on ℝ. Consequently, all ordered two-by-two minors of the translation kernel K(x−y) are strictly positive. The proof uses an explicit formula in exponentials and trigonometric functions, followed by elementary estimates on a compact interval and on the tail.

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