Geodesic
Representing non-periodic functions of bounded $\Lambda$-variation by multi-dimensional Fourier integrals
Izvestiya. Mathematics , Tome 67 (2003) no. 6, pp. 1081-1100

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Sufficient conditions are established for the convergence of the multiple Fourier integral of a Pringsheim integrable function (in the sense of convergence of partial integrals over parallelepipeds) in terms of the membership of the function in classes of bounded $\Lambda$-variation. These conditions require the following: the function should belong to a class of bounded harmonic variation, the point under consideration should be regular, the harmonic variation should behave “well” in the neighbourhood of the point, and the function should be continuous with respect to the harmonic variation on a special subset in the neighbourhood of infinity. It is also shown that, in general, neither of the last two conditions can be dropped. 
@article{IM2_2003_67_6_a0,
     author = {A. N. Bakhvalov},
     title = {Representing non-periodic functions of bounded $\Lambda$-variation by multi-dimensional {Fourier} integrals},
     journal = {Izvestiya. Mathematics },
     pages = {1081--1100},
     publisher = {mathdoc},
     volume = {67},
     number = {6},
     year = {2003},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/IM2_2003_67_6_a0/}
}
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A. N. Bakhvalov. Representing non-periodic functions of bounded $\Lambda$-variation by multi-dimensional Fourier integrals. Izvestiya. Mathematics , Tome 67 (2003) no. 6, pp. 1081-1100. http://geodesic.mathdoc.fr/item/IM2_2003_67_6_a0/