Abstract:
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,
Lebesgue points, Wiener approximation theorem.
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