Fourier Duality on Noncompact Symmetric Spaces: A Paley–Wiener Perspective — E8 Intelligence Research

FINDING: The search yields no direct categorical Fourier transform level duality or affine Grassmannian result; instead, it surfaces standard Fourier pedagogy and one advanced paper on Paley–Wiener theory for joint-eigenspace Fourier transforms on noncompact symmetric spaces. | MATH: No explicit equations or constants extracted from the listed sources; the arxiv paper (2409.09036v1) concerns Poisson transform, Helgason Fourier transform, and duality on symmetric spaces of noncompact type — but no specific formula, ratio, or constant is given in the search snippet. | CONNECTION: The symmetric space duality (noncompact type) is tied to root systems and Weyl groups, which underpin crystallographic symmetries (e.g., A_n, B_n, C_n, D_n, E_6, E_7, E_8). However, no golden ratio, reciprocal pair, or base-60 structure is evidenced in the provided text. The Fourier duality property (time↔frequency) is a linear involution, not a geometric ratio. | DEPTH: 2 — The finding is mostly generic Fourier Author: Andrew Stewart Caldin, Independent Researcher, UK. Part of the E8 Intelligence Research series. Platform: e8intelligence.com