# Fourier series normalize

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.

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?

• 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 $[-\pi,\pi]$. – GPerez Jun 1 '15 at 22:55

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 $e^{-\mu\pi}$.