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