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

Voir la notice de l'article provenant de la source Math-Net.Ru

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, Wiener approximation theorem.
Mots-clés : Marcinkiewicz sums of a double Fourier series, Lebesgue points 
@article{SM_2018_209_5_a6,
     author = {R. M. Trigub},
     title = {The {Fourier} transform of bivariate functions that depend only on the maximum of the absolute values of their variables},
     journal = {Sbornik. Mathematics},
     pages = {759--779},
     publisher = {mathdoc},
     volume = {209},
     number = {5},
     year = {2018},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_2018_209_5_a6/}
}
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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/