Find a sequence of continuous periodic functions which converge in $L^2$ to a discontinuous periodic function. (Hint- Try converging to the square wave function)
Answer:
If $\{f_n(x) \}_n$ be the sequence of continuous periodic function converging to a discontinuous periodic function $f(x)$ in $L^2$, then $$ \lim_{n \to \infty} \int_{0}^{2 \pi} |f_n(x)-f(x)|^2dx =0.$$
We have to show the above.
The hints says, try converging to the square wave function.
An example of square wave function is
$f(x)=\text{sgn} \left( \sin \frac{2 \pi}{T}\right)$, where $\text{sgn}$ denote the $\text{sign}$ funnction , i.e, $\pm 1$ and $T$ is the period.
Now how to use this ?