A Geometric Fourier Extension: Spectral Calculus for Nonlinear and Non-Markovian Dynamics

This framework draft introduces the Geometric Fourier Extension (GFE), a rigorous functional-analytic extension of the classical Fourier transform to nonlinear, nonlocal (non-Markovian) evolution equations of the form∂_t u + N[u] = ∫ K(t-s) L[u(s)] ds + f(t).The approach combines two key innovations:A geometric spectral transform (GST): fields are expanded in the eigenbasis of a field-dependent Laplace–Beltrami operator −Δ_{g[u]}, providing an adaptive, nonlinear spatial basis that diagonalizes differential structure for evolving nonlinear geometry.A provable sum-of-exponentials (SOE) embedding: causal memory kernels are approximated by finite positive-weight decaying exponentials, converting nonlocal convolutions into local auxiliary ODEs on an extended state space.Under suitable regularity and approximability assumptions, we prove local/global well-posedness, uniqueness, stability, and uniform approximation of solutions (error controlled by SOE L¹-norm). We derive extended energy/Hamiltonian identities for dissipative systems and discuss spectral links to fractional dynamics. Applications include viscous Burgers with memory, nonlinear Schrödinger with nonlocal damping, Nakajima–Zwanzig equation ,weak-field fractional general-relativistic n-body problems. Practical numerical recipes (SOE fitting via vector fitting/NNLS, IMEX integration, complexity estimates) are provided.This work offers a systematic generalized spectral calculus for problems combining nonlinearity and temporal memory, previously challenging due to loss of diagonalization. It lays groundwork for further developments in non-Markovian PDEs and related operators.