# Fast logarithmic Fourier-Laplace transform of nonintegrable functions

## TL;DR

This paper introduces a fast, flexible numerical Fourier-Laplace transform method that efficiently handles nonintegrable functions with power-law tails, demonstrating exponential convergence and broad applicability in physical models.

## Contribution

It extends the logarithmic Fourier transform to nonintegrable functions, providing a method with proven exponential convergence and practical guidelines for applications involving power-law tails.

## Key findings

- Achieves exponential convergence for a broad class of functions.
- Effectively handles functions with power-law asymptotic behavior.
- Demonstrates applicability in physical models near criticality.

## Abstract

We present an efficient and very flexible numerical fast Fourier-Laplace transform, that extends the logarithmic Fourier transform (LFT) introduced by Haines and Jones [Geophys. J. Int. 92(1):171 (1988)] for functions varying over many scales to nonintegrable functions. In particular, these include cases of the asymptotic form $f(\nu\to0)\sim\nu^a$ and $f(|\nu|\to\infty)\sim\nu^b$ with arbitrary real $a>b$. Furthermore, we prove that the numerical transform converges exponentially fast in the number of data points, provided that the function is analytic in a cone $|\Im{\nu}|<\theta|\Re{\nu}|$ with a finite opening angle $\theta$ around the real axis and satisfies $|f(\nu)f(1/\nu)|<\nu^c$ as $\nu\to 0$ with a positive constant $c$, which is the case for the class of functions with power-law tails. Based on these properties we derive ideal transformation parameters and discuss how the logarithmic Fourier transform can be applied to convolutions. The ability of the logarithmic Fourier transform to perform these operations on multiscale (non-integrable) functions with power-law tails with exponentially small errors makes it the method of choice for many physical applications, which we demonstrate on typical examples. These include benchmarks against known analytical results inaccessible to other numerical methods, as well as physical models near criticality.