On the Fourier transform of function of two variables which depend only on the maximum of these variables

TL;DR

This paper characterizes when the Fourier transform of functions depending only on the maximum of two variables is integrable, providing necessary and sufficient conditions, and relates positivity of the Fourier transform to a one-dimensional problem.

Contribution

It offers a complete characterization of the Fourier transform's integrability for max-dependent functions and reduces the positivity question to a one-dimensional analysis.

Findings

Derived necessary and sufficient conditions for Fourier transform integrability.

Reduced positivity of Fourier transform to a one-dimensional problem.

Established criteria involving the function's behavior at infinity.

Abstract

For functions $f(x_{1},x_{2})=f_{0}\big(\max\{|x_{1}|,|x_{2}|\}\big)$ from $L_{1}(\mathbb{R}^{2})$, sufficient and necessary conditions for the belonging of their Fourier transform $\widehat{f}$ to $L_{1}(\mathbb{R}^{2})$ as well as of a function $t\cdot \sup\limits_{y_{1}^{2}+y_{2}^{2}\geq t^{2}}\big|\widehat{f}(y_{1},y_{2})\big|$ to $L_{1}(\mathbb{R}^{1}_{+})$. As for the positivity of $\widehat{f}$ on $\mathbb{R}^{2}$, it is completely reduced to the same question on $\mathbb{R}^{1}$ for a function $f_{1}(x)=|x|f_{0}\big(|x|\big)+\int\limits_{|x|}^{\infty}f_{0}(t)dt$.