Almost everywhere convergence of multiple Fourier series of monotonic functions

M. I. Dyachenko

Sbornik. Mathematics, Tome 73 (1992) no. 1, pp. 11-25

Résumé

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 $(\mathbf a,\mathbf b)$.