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

Abstract

For functions $f(x_1,x_2)=f_0(\max \{|x_1|,|x_2|\})$ from $L_1({\mathbb{R}}^2)$, sufficient and necessary conditions for the belonging of their Fourier transform $\hat{f}$ to $L_1({\mathbb{R}}^2)$ as well as of a function $t\cdot \underset{y_1^2+y_2^2\ge t^2}{\sup }|\hat{f}(y_1,y_2)|$ to $L_1({\mathbb{R}}_{+}^1)$. As for the positivity of $\hat{f}$ on ${\mathbb{R}}^2$, it is completely reduced to the same question on ${\mathbb{R}}^1$ for a function $f_1(x)=|x|f_0(|x|)+\underset{|x|}{\overset{\infty }{\int }}{f}_0(t)dt$.

30 pages; the paper is in Russian, with the title, abstract and key words translated