Functions and Analysis

The Inherent Mixed-Radix Structure of FFT: A General Framework for Puiseux Series and Branch Cut Computation

Authors: Chang Hee Kim
Comments: 8 Pages. Licensed under CC BY-NC-ND 4.0

This paper presents a new foundational insight into the Fast Fourier Transform (FFT) algorithm: mixed-radix decomposition is not an optional design, but an inherent structural property of FFT itself. We demonstrate that any power series f(x), when decomposed into modulo-s subseries for any integer s ≥ 2, naturally aligns with radix-s FFT computation — regardless of whether s is prime. This unification renders the need for specialized mixed-radix FFT frameworks obsolete. Furthermore, we show that this structure enables seamless extraction of Puiseux series coefficients from a function, including those involving fractional powers and branch cuts. The FFT not only resolves monodromy behavior computationally, but also simplifies the treatment of multivalued functions across Riemann surfaces without symbolic intervention. This paper is a direct sequel to the author’s previous work, Sampling on the RiemannSurface: A Natural Resolution of Branch Cuts in Puiseux Series, and the first foundational paper, A Unified Computational Framework Unifying Taylor-Laurent, Puiseux, Fourier Series, and theFFT Algorithm.
Category: Functions and Analysis