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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. –  Simon Markett Jun 11 '12 at 11:33
    
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. –  user31373 Jun 11 '12 at 14:33
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A trigonometric series that converges uniformly on $[-\pi,\pi]$ also converges uniformly on all of $\mathbb R$. Its sum is obviously a $2\pi$ periodic function. Therefore, the answer is yes: if $f$ is the sum of a uniformly convergent trigonometric series, then so is the $2\pi$-periodic extension of $f$.

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