Spectral Mollification of Singular Distributions: A Latent Framework Approach

We present a framework for mollifying singular probability distributions into smooth spectral representations via the Latent transform. The key result is a convergence theorem showing that any distribution with finite second moment can be approximated to arbitrary precision by a truncated spectral expansion. Under a super-decay assumption on the spectral coefficients, the rate of convergence is \(O(N^{-2})\) in \(L^2\) norm for distributions with bounded variation. Numerical experiments suggest a faster \(O(N^{-4})\) rate for absolutely continuous distributions. We demonstrate applications to financial risk modeling and physics simulation.