Fourier transform of composed functions

TL;DR

This paper derives an explicit formula for the Fourier transform of composed functions, enabling easier computation of transforms for complex functions, demonstrated through a specific example involving hyperbolic functions.

Contribution

It provides a new explicit formula for the Fourier transform of composed functions, expanding analytical tools for Fourier analysis.

Findings

Derived an explicit Fourier transform formula for composed functions.

Applied the formula to compute the transform of a hyperbolic function.

Demonstrated the formula's usefulness with a concrete example.

Abstract

We prove an explicit formula for the Fourier transform of $f(u(t))$, given the Fourier transform of $f(t)$, assuming $f\in L^2(-\infty,\infty)$ and $u$ sufficiently well behaved. We illustrate its usefulness by calculating the Fourier transform of $(a^2 + \sinh(bt)^2)^{-1}$.