Geodesic
The Fourier transform of bivariate functions that depend only on the maximum of the absolute values of their variables
Sbornik. Mathematics, Tome 209 (2018) no. 5, pp. 759-779

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Given an $L_1(\mathbb{R}^2)$-function $f(x_1,x_2)=f_0(\max\{|x_1|,|x_2|\})$, necessary conditions and sufficient conditions for its Fourier transform $\widehat{f}$ to lie in $L_1(\mathbb{R}^2)$ and for the function $t\mapsto t\sup_{y_1^2+y_2^2\geqslant t^2}|\widehat{f}(y_1,y_2)|$ to be in $L_1(\mathbb{R}_{+})$ are indicated. The problem of the positivity of $\widehat{f}$ on $\mathbb{R}^2$ is shown to be completely reducible to the same problem for the function $\displaystyle f_1(x)=|x|f_0(x)+\int_{|x|}^\infty f_0(t)\,dt$ in $\mathbb{R}$. Bibliography: 20 titles.
Keywords: Wiener Banach algebra, positive definiteness, Bernstein's theorem on completely monotone functions, Marcinkiewicz sums of a double Fourier series, Wiener approximation theorem.
Mots-clés : Lebesgue points 
R. M. Trigub. The Fourier transform of bivariate functions that depend only on the maximum of the absolute values of their variables. Sbornik. Mathematics, Tome 209 (2018) no. 5, pp. 759-779. http://geodesic.mathdoc.fr/item/SM_2018_209_5_a6/
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