Positive-Definite Boundary Correlations and Fourier Zero Strips: Resolution of Csordas' Open Problem 4.12

Abstract

This paper gives a necessary and sufficient positive-definiteness criterion for confining the zeros of Fourier transforms of real even kernels with super-Gaussian decay to a prescribed horizontal strip. The criterion is formulated in terms of explicit boundary correlation kernels and applies without any restriction on the number of nonreal zeros. The main theorem also provides a boundary-jet characterization, a criterion using only a countable dense set of heights, and an exact method for recovering the minimal width of the zero strip from the associated correlation family. Specialization to admissible kernels resolves George Csordas’ Open Problem 4.12. Together with the author’s earlier resolutions of Open Problems 4.10, 4.11, 4.13, and 4.14, this result completes Csordas’ sequence of Open Problems 4.10–4.14. Research methodology and AI assistance:This work was developed using the CARMA-Math research workflow, a cumulative AI-assisted mathematical research methodology using persistent research archives, literature and prior-art investigation, iterative proof exploration, and verification procedures. Generative AI (ChatGPT) was used extensively for mathematical exploration, proof development, computational reasoning, literature research, and manuscript preparation.