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Let $f(x)$ be the value of a trigonometric series, which converges uniformly on $\left[ -\pi, \pi\right]$. If I multiply $f(x)$ with $e^{iax}$ where $a\in\mathbb{N}$ will the result then be a trigonometric series which converges uniformly?

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    a is$a$constant.2012-06-11

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Let $f_n$ denote the n-th partial sum of the series. We have $f_n \to f$ uniformly on $[-\pi,\pi].$ Clearly, $e^{iax} f_n(x) \to e^{iax} f(x)$ pointwise, and this is uniform since $ \sup_{x\in [-\pi,\pi]} \| e^{iax}f_n(x) - e^{iax}f(x)\| =\sup_{x\in [-\pi,\pi]} \| f_n(x) - f(x)\| \to 0 \text{ as } n\to \infty.$

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    @characters By the usual definition of$a$trigonometric series, yes ($a$ must be an integer).2012-06-11