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Let $ \mu \in \mathbb { R} $ and let

$$ f ( x ) = e ^{ \mu x } ,\ x \in (- \pi , \pi ] . $$

i) Arguments for that the Fourier series $ \sum_ { k = - \infty } ^ \infty c_k e ^ { ikx } $ for $ f $ converges pointwise on $ \mathbb { R} $ and

ii) compute the sum function .

iii)Decide also whether there are uniform convergence.

Answer:

i) The function $ f $ is piecewise differentiable , So is the Fourier series pointwise convergent,

ii) i have a problem how to normalize f, i can see that the function is discontinuous in $\pi$ so what to do?

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  • $\begingroup$ Note that you can go from $L_2((-\pi,\pi])$ to $L_2([-\pi,\pi])$ via $f(-\pi):=e^{-\mu\pi}$ (if this was your concern). Otherwise I don't know what you mean by "discontinuous"; $f$ is certainly continuous in $(-\pi,\pi]$ and in $[-\pi,\pi]$. $\endgroup$ – GPerez Jun 1 '15 at 22:55
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The periodic extension of $f$ to the real line $\mathbb{R}$ should be considered here. This extension is continuous everywhere except at the points $(2k+1)\pi$ with $k\in\mathbb{Z}$, where the left limit is $e^{\mu\pi}$ and the right limit is $e^{-\mu\pi}$.

Since this is a piecewise differentiable function which has only jump discontinuities, the Fourier series will converge pointwise to it, except that at the discontinuity points it will converge to the average of left and right limits.

The convergence is not uniform, because of those discontinuities.

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