On nearly radial product functions

TL;DR

This paper proves that if the product of a function with itself is nearly radially symmetric, then the original function is close to a Gaussian, with a precise quantitative measure of this closeness.

Contribution

It establishes a quantitative stability result linking near radial symmetry of a product function to the Gaussian nature of the original function, with optimal exponent.

Findings

Near radial symmetry of $f(x)f(y)$ implies $f$ is close to a Gaussian

Provides a quantitative stability estimate with optimal exponent

Extends understanding of symmetry and stability in functional analysis

Abstract

If $f\in L^2(R^d)$ and if the function $f(x)f(y)$ is close in $L^2(R^{2d})$ norm to a radially symmetric function of $(x,y)$ then $f$ is close in $L^2$ norm to a centered Gaussian function. This is proved in a quantitative form with the optimal exponent measuring closeness.