I have given a set of functions in $L^2\left(\left[-\frac{a}{2}, -\frac{a}{2} \right]\right)$ consisting of the following functions:
$$u_{n}(x)=\sqrt{\frac{2}{a}}f_n(x),$$
where
$f_n(x)= \sin\left(\frac{n\pi}{a}x\right)$, if $n\in\mathbb{N}$ is even and
$f_n(x)=\cos\left(\frac{n\pi}{a}x\right)$, if $n\in\mathbb{N}$ is odd.
I want to prove that this set of functions form an orthonormal basis for $L^2$. I could show that these functions form an orthonormal set but couldn't prove that it is actually an orthonormal basis.
Do you have any idea how I can proceed in this situation? For instance, how can I approximate simple polynomial like $p(x)=x^d$ for $d\in\mathbb{N}$ with respect to the norm $\Vert\cdot\Vert_{L^2}$?
I would be very happy if someone can help me. Best regards.
These functions all vanish at the endpoints $x=\pm \frac{a}{2}$. So the equivalent problem is to deal with $\{ \sin\left(\frac{n\pi x}{a}\right)\}_{n=1}^{\infty}$ on $[0,a]$. You can see that this is true because $\sin(\frac{n\pi(x+a/2)}{a})=\sin(\frac{nx}{a}+\frac{n\pi}{2})$ is $\pm\cos(\frac{nx}{a})$ if $n$ is odd and is $\pm\sin(\frac{nx}{a})$ if $n$ is even. Your problem then reduces the Fourier sine series on $[0,a]$. The Fourier sine series of $f$ on $[0,a]$ is the same as the classic Fourier expansion of the odd extension $f_{o}$ on $[-a,a]$. So, through a couple of steps, everything reduces to the classical Fourier series.