Abstract It is shown that any function acting from the real line ℝ {\mathbb{R}} into itself can be expressed as a pointwise limit of finite sums of periodic functions. At the same time, the real analytic function x → exp ⁡ ( x 2 ) {x\rightarrow\exp(x^{2})} cannot be represented as a uniform limit of finite sums of periodic functions and, simultaneously, this function is a locally uniform limit of finite sums of periodic functions. The latter fact needs the techniques of Hamel bases.

中文翻译：

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关于周期函数的有限和

摘要 证明了从实线ℝ {\mathbb{R}} 作用于自身的任何函数都可以表示为周期函数的有限和的逐点极限。同时，实解析函数 x → exp ⁡ ( x 2 ) {x\rightarrow\exp(x^{2})} 不能表示为周期函数有限和的统一极限，同时，这个函数是周期函数有限和的局部一致极限。后一个事实需要 Hamel 碱基技术。