Geodesic
Equiconvergence of expansions in a~multiple Fourier series and Fourier integral for summation over squares
Izvestiya. Mathematics , Tome 10 (1976) no. 3, pp. 652-671

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In this work there are constructed a function $f(\overline x)\in L_1([-\pi,\pi]^2)$ such that the difference between the Fourier series expansion and the Fourier integral expansion for summation over squares diverges almost everywhere on $\{[-\pi,\pi]^2\}$, and a function $f(\overline x)\in L_p([-\pi,\pi]^N)$, $p>1$, $N\geqslant2$, for which the difference diverges at a point. Bibliography: 5 titles. 
@article{IM2_1976_10_3_a9,
     author = {I. L. Bloshanskii},
     title = {Equiconvergence of expansions in a~multiple {Fourier} series and {Fourier} integral for summation over squares},
     journal = {Izvestiya. Mathematics },
     pages = {652--671},
     publisher = {mathdoc},
     volume = {10},
     number = {3},
     year = {1976},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_1976_10_3_a9/}
}
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I. L. Bloshanskii. Equiconvergence of expansions in a~multiple Fourier series and Fourier integral for summation over squares. Izvestiya. Mathematics , Tome 10 (1976) no. 3, pp. 652-671. http://geodesic.mathdoc.fr/item/IM2_1976_10_3_a9/