In this paper we refine the re-expansion problems for the one-dimensional torus and extend them to the multidimensional tori and to compact Lie groups. First, we establish weighted versions of classical re-expansion results in the setting of multi-dimensional tori. A natural extension of the classical re-expansion problem to general compact Lie groups can be formulated as follows: given a function on the maximal torus of a compact Lie group, what conditions on its (toroidal) Fourier coefficients are sufficient in order to have that the group Fourier coefficients of its central extension are summable. We derive the necessary and sufficient conditions for the above property to hold in terms of the root system of the group. Consequently, we show how this problem leads to the re-expansions of even/odd functions on compact Lie groups, giving a necessary and sufficient condition in terms of the discrete Hilbert transform and the root system. In the model case of the group \(\mathrm{SU(2)}\) a simple sufficient condition is given.

中文翻译：

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紧凑型李群的重新展开

在本文中，我们对一维环面的重新展开问题进行了细化，并将其扩展到多维环面和紧致李群。首先，我们在多维花托的设置中建立经典再扩展结果的加权版本。可以将经典再扩展问题自然扩展到一般紧致Lie基团，可以表示为：给定紧致Lie基团的最大环面函数，对其（环）傅立叶系数有什么条件足以使其具有其中心扩展的群傅立叶系数可求和。我们根据组的根系统得出上述属性必须具备的必要条件和充分条件。因此，我们展示了这个问题如何导致紧凑的Lie群上的偶/奇函数的重新展开，在离散Hilbert变换和根系统方面给出了充要条件。在小组的情况下\（\ mathrm {SU（2）} \）给出了一个简单的充分条件。