In this paper we construct an integrable function of two variables for which the double Fourier-Walsh series converges both by rectangles and by spheres. Besides, we show that the coefficients of the series on the spectrum are positive and are arranged in decreasing order in all directions. Also, it is proved that after a suitable choice of signs for the Fourier coefficients of the series the spherical partial sums of the obtained series are dense in L^p[0, 1]², p ∈ (0, 1).

中文翻译：

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双通用付里叶系列

在本文中，我们构造了两个变量的可积函数，双傅里叶-沃尔什级数均通过矩形和球面收敛。此外，我们表明该谱系列的系数是正的，并且在所有方向上都以降序排列。另外，还证明了标志为系列的傅立叶系数的适当选择后获得的一系列的球形部分和是在密集的大号^p [0，1] ²，p ∈（0，1）。