We investigate analytic properties of the double Fourier sphere (DFS) method, which transforms a function defined on the two-dimensional sphere to a function defined on the two-dimensional torus. Then the resulting function can be written as a Fourier series yielding an approximation of the original function. We show that the DFS method preserves smoothness: it continuously maps spherical H\"older spaces into the respective spaces on the torus, but it does not preserve spherical Sobolev spaces in the same manner. Furthermore, we prove sufficient conditions for the absolute convergence of the resulting series expansion on the sphere as well as results on the speed of convergence.

翻译：我们调查了双Fourier球体(DFS)的分析特性,该方法将二维球体上定义的函数转换成二维象体上定义的函数。然后,由此产生的函数可以写成产生原始函数近似值的Fourier序列。我们显示,DF方法保持了光滑性:它不断将球体H\"老化器空间绘制成横线上各自的空间,但并没有以同样的方式保护球体索博列空间。此外,我们证明,我们有足够的条件,可以将由此形成的球体扩展序列绝对合并到球体上,并取得汇合速度的结果。