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

R. M. Trigub

Published 2015-12-10Version 1

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 $f_{1}(x)=|x|f_{0}\big(|x|\big)+\int\limits_{|x|}^{\infty}f_{0}(t)dt$.