Geodesic
Almost everywhere convergence of multiple Fourier series of monotonic functions
Sbornik. Mathematics, Tome 73 (1992) no. 1, pp. 11-25 Cet article a éte moissonné depuis la source Math-Net.Ru

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Let $m$ be a natural number, $m\geqslant2$. Then we shall say that a function $f(\mathbf t)$ of period $2\pi$ in each variable is monotonic if there exist an open rectangular parallelepiped $(\mathbf a,\mathbf b)=\prod\limits_{j=1}^m(a_j,b_j)\subseteq [-\pi,\pi)^m$ and numbers $\gamma_1,\dots,\gamma_m$, each of which is either 0 or 1, such that $f(\mathbf t)=0$ for $\mathbf t\in [-\pi,\pi)^m\setminus(\mathbf a,\mathbf b)$, and if $\mathbf x,\mathbf y\in (\mathbf a,\mathbf b)$ and $(-1)^{\gamma_j}x_j\leqslant(-1)^{\gamma_j}y_j$ for $j=1,\dots,m$, then $f(\mathbf x)\geqslant f(\mathbf y)$. The main result of this paper is that the multiple trigonometric Fourier series of an integrable monotonic function is Pringsheim convergent almost everywhere, in particular at each point of continuity of $f(\mathbf t)$ in the interior of $(\mathbf a,\mathbf b)$. 
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     author = {M. I. Dyachenko},
     title = {Almost everywhere convergence of multiple {Fourier} series of monotonic functions},
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     year = {1992},
     volume = {73},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1992_73_1_a1/}
}
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M. I. Dyachenko. Almost everywhere convergence of multiple Fourier series of monotonic functions. Sbornik. Mathematics, Tome 73 (1992) no. 1, pp. 11-25. http://geodesic.mathdoc.fr/item/SM_1992_73_1_a1/

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