Fourier Analysis on Twisted Spaces, An introduction to the Möbius and Klein transforms — lecture notes for university students

Abstract

An introductory set of notes for university students (BSc/MSc) on the Möbius and Kleintransforms. The starting point is an everyday phenomenon: many engineering and physical quantities are notperiodic but antiperiodic — after half a period they go over into their negative (half-wavesymmetric signals, orientation fields, spinors). The ordinary Fourier series then leaks: the spectrum of asingle pure oscillation is smeared over all harmonics. The notes derive the basis suited to such signals —the half-integer frequency exponentials — establish the sector decomposition theorem(f_=12(f() f(+2)), the eigenspaces of asingle unitary involution), give the discrete version in five lines of NumPy (pre-modulation + FFT,O(M M)), and then the Klein transform by symmetrizing on thecovering torus, with the pairing rule (cosine ↔ integer, sine ↔ half-integer frequency). Four figures,several worked examples, check-yourself code snippets and seven exercises support understanding; a separatebox states what the transform does not do. The complete treatment with proofs and numericalverification is in the companion research article.