Integral Transforms with Finite-Principal-Part Laurent Kernels: Weighted Dilation Operators, Mellin Symbols, and Exact Signal Inversion

We extend the master-integral-transform theory from entire kernels to finite-principal-part Laurent kernels and show that the resulting transform is a weighted dilation operator acting on the Fourier transform of a weighted signal. This yields a unified operator framework for several exact inversion mechanisms, including Mellin diagonalization, two-sided Mellin-symbol inversion, Dirichlet–Wiener inversion, log-scale Fourier inversion, recursive inversion, and Neumann-series recovery. The main structural result is that finite negative Laurent tails do not destroy the spectral architecture; they enlarge the one-sided dilation orbit to a two-sided one. We establish exact factorization formulas on weighted function spaces, prove branchwise Mellin inversion under explicit integrability assumptions, derive a contour-free Dirichlet–Wiener inverse, obtain a log-scale Fourier multiplier representation suitable for FFT-based recovery, and prove a practical stability bound away from multiplier zeros. A worked symbolic example and a numerical blueprint are also included.