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. 
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     title = {Equiconvergence of expansions in a~multiple {Fourier} series and {Fourier} integral for summation over squares},
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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/

[1] Bloshanskii I. L., “O ravnoskhodimosti razlozhenii v kratnyi trigonometricheskii ryad Fure i integral Fure”, Matem. zametki, 18:2 (1975), 153–168

[2] Bari N. K., Trigonometricheskie ryady, Fizmatgiz, M., 1961 | MR

[3] Zhak I. E., “O sopryazhennykh dvoinykh trigonometricheskikh ryadakh”, Matem. sb., 31:3 (1952), 469–484

[4] Tevzadze N. R., “O skhodimosti dvoinogo ryada Fure funktsii, summiruemoi s kvadratom”, Soobsch. AN GruzSSR, 58:2 (1970), 277–279 | MR | Zbl

[5] Sjolin P., “Convergence almost every where of certain singular integrals and multiple Fourier series”, Arkiw Math., 9:1 (1971), 65–90 | DOI | MR