On the Wiener-Khinchin transform of functions that behave as approximate power-laws. Applications to fluid turbulence

TL;DR

This paper studies the stability and properties of the Fourier transform of functions resembling power-laws, with applications to fluid turbulence, providing error estimates and criteria for approximate homogeneity.

Contribution

It introduces refined stability results for Fourier transforms of approximate power-law functions and connects these findings to turbulence theory.

Findings

Provides error estimates for Fourier transforms outside the homogeneous range

Establishes criteria for approximate homogeneity near zero and infinity

Numerical simulations confirm the sharpness of theoretical results

Abstract

As we all know, the Fourier transform is continuous in the weak sense of tempered distribution; this ensures the weak stability of Fourier pairs. This article investigates a stronger form of stability of the pair of homogeneous profiles $(|x|^{-\alpha}, c_d |\xi|^{d-\alpha})$ on $\mathbb{R}^d$. It encompasses, for example, the case where the homogeneous profiles exist only on a large but finite range. In this case, we provide precise error estimates in terms of the size of the tails outside the homogeneous range. We also prove a series of refined properties of the Fourier transform on related questions including criteria that ensure an approximate homogeneous behavior asymptotically near the origin or at infinity. The sharpness of our results is checked with numerical simulations. We also investigate how these results consolidate the mathematical foundations of turbulence theory.