The Möbius–Klein–Laplace Transformation: The MKT in the Complex Frequency Plane
Year: 2026
Type: preprint
Abstract: The Möbius–Klein transform (MKT) is a Fourier analysis on twisted spaces. It has no damping, no starting time and no initial values. This article takes the classical steps from the Fourier transform to the Laplace transform on the twisted sector. The period is kept as a seam, and the damping is added on top of it. The result is the Möbius–Klein–Laplace transformation (MKLT): the Laplace transform of the seam-closed signal, in which every part of a causal signal returns at each seam with a complex seam parameter. Zero gives the Laplace transform, minus one the Möbius seam, a unimodular value a Bloch twist, a smaller modulus a lossy seam, and a reflection operator the Klein bottle. The MKLT keeps the whole Laplace calculus. The initial value returns at every seam, a damping factor renormalises the seam parameter, and the convolution theorem keeps one seam factor. A nonzero history of a seamed system enters as an equivalent input on the first cell. The final-value theorem has 2 successors: a limit for a lossy seam, and a mean for every unimodular seam other than the periodic seam. The inverse splits a response into a transient and a steady state on the seam row, and the steady state of the Möbius seam is exactly the MKT series of the twisted periodisation. The Laplace transform is recovered in 3 ways: at the zero seam, as the period tends to infinity, and as the average over the seam circle. The discrete MKLT does the same for the z-transform and is the calculus of the feedback comb. The inverse MKLT is a Bromwich inversion followed by an unsealing with one period of memory. Its unsealing step needs no condition on the seam parameter and has a bounded gain, and it gives estimators of the seam parameter and period. Read as a loop, the modulus of the seam parameter sets the damping of the seam row and its argument shifts the row. An imaginary seam gives a quarter row, a complex seam on a real signal needs an I/Q pair, a filter in the return path curves the row, and a noise-driven loop has the Airy spectrum of a Fabry–Pérot resonator. A periodically changing seam acts through its monodromy, and the inverse stays local for every changing seam. An operator seam carries the construction to non-orientable spaces. On the Klein bottle the seam is a reflection whose sectors choose the pole rows. Across a reflected fibre the bilateral transform needs a symmetric strip, and the parity parts admit a one-sided one. On the flat 3- and 4-manifolds glued along one direction by a reflection, on the spin structures of the torus and on the solid Klein bottle the MKLT gives sector sum rules and sector-wise decay rates. In a Floquet coordinate of a periodic system the response is the seam closure of its one-period response, which places period doubling inside the same calculus. Applications cover the quarter-wave resonator and lossy or reactive transmission lines, reflection matrices and mirrored round trips, comb filters and odd-harmonic repetitive control, half-wave symmetric drives, neutral delay systems, echo and loop removal with moving-target cancellers and dereverberation, period doubling, fermionic Matsubara frequencies, twisted heat flow, complex Bloch waves and mirrored ring cavities. Differential equations with a seam are solved for quasi-periodic boundary conditions, neutral delays, Hill equations and reflecting lines, and flat bundles give their geometric reading. Worked examples apply the transform and its inverse. Every new statement is proved and checked by deterministic scripts with counter-checks.
Author: Mark, Laszlo
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7225585569