Geodesic
About the convergence of the Fourier transform
Daghestan Electronic Mathematical Reports, Tome 15 (2021), pp. 1-21

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The main result is the proof of the theorems, the results of which one can characterize as a weak form of the formula for the inversion of the bi-dimmensional Fourier transform. Sufficient conditions on a function are obtained for a weak (of degree $r$) convergence of bi-dimmensional Fourier transform for a function $f(x;y)$. These conditions have an integral form and describe the behavior of the function near the border of a rectangle. A similar theorem is proved, in which the Fourier transform of a function $f$ is replaced by the Fourier transform of another function $g$, the norm of the central difference of which does not exceed the norm of the central difference of $f$. The principal objective is to study the behavior of the Fourier transform of $g$ and $f$.
Keywords: two-dimensional Fourier transform, Riemann-Lebesgue theorem. 
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M. A. Boudref. About the convergence of the Fourier transform. Daghestan Electronic Mathematical Reports, Tome 15 (2021), pp. 1-21. http://geodesic.mathdoc.fr/item/DEMR_2021_15_a0/