In this note, we investigate the partial Fourier series on a product of two compact Lie groups. We give necessary and sufficient conditions for a sequence of partial Fourier coefficients to define a smooth function or a distribution. As an application, we will study conditions for the global solvability of an evolution equation defined on ${\mathbb{T}}^1\times {\mathbb{S}}^3$ and we will show that some properties of this evolution equation can be obtained from a conjugated constant-coefficient equation.

中文翻译：

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紧凑李群的部分傅里叶级数

在本说明中，我们研究了两个紧致李群的乘积的部分傅里叶级数。我们给出了部分傅里叶系数序列的必要和充分条件，以定义平滑函数或分布。作为一种应用，我们将研究定义在上的演化方程的整体可解性的条件${\mathbb{Ť}}^{\mathrm{1个}}\times {\mathbb{小号}}^3$ 并且我们将证明，该演化方程的某些性质可以从共轭常数系数方程获得。