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Mat. Sb., 2018, Volume 209, Number 5, Pages 166–186 (Mi msb8888)  

The Fourier transform of bivariate functions that depend only on the maximum of the absolute values of their variables

R. M. Trigub

Sumy State University, Ukraine

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 $t\mapsto t\sup_{y_1^2+y_2^2\geq t^2}|\widehat{f}(y_1,y_2)|$ 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.

DOI: https://doi.org/10.4213/sm8888

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English version:
Sbornik: Mathematics, 2018, 209:5, 759–779

Bibliographic databases:

UDC: 517.518.5+517.518.476
MSC: Primary 42B10; Secondary 42B35
Received: 18.12.2016 and 03.05.2017

Citation: R. M. Trigub, “The Fourier transform of bivariate functions that depend only on the maximum of the absolute values of their variables”, Mat. Sb., 209:5 (2018), 166–186; Sb. Math., 209:5 (2018), 759–779

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