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If $f:[-\pi, \pi] \rightarrow\mathbb{C}$ is the value of an uniform convergent trigonometric series, can I then deduce that the $2\pi$-periodic normalized extension is an uniform convergent trigonometric series?

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    I am not sure whether the notation you use is totally standard. Well, even if it is I am pretty sure that people (like me) could answer your question if you made clearer what you mean by every word.2012-06-11
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    The following is true: if a series $\sum_n (a_n \cos nx+b_n\sin nx)$ converges uniformly on $[-\pi,\pi]$, then it converges uniformly on $\mathbb R$. Whether or not this answers your question I can only guess.2012-06-11

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