I know that for a real-valued function set $\{f_n(x)\}$, its completeness condition is $\Sigma_n f_n(x)=\delta(x-x')$. That is, this condition guarantees that a well-behaved function can be write as a series of addition of $\{f_n(x)\}$.
Is there a similar condition for function of period $2\pi$?
For example, I know $\text{sin}(nx)$ and $\text{cos}(nx)$ can expand any well-behaved function of period $2\pi$. What condition is fulfilled by such a function set?
Let $f$ be a $2\pi$ periodic function on $[-\pi,\pi]$. The truncated Fourier series for $f$ which includes terms $1,\cos(nx),\sin(nx)$ for $1 \ne n \le N$ can be written as the Dirichlet integral $$ \frac{1}{\pi}\int_{-\pi}^{\pi} f(t)\frac{\sin((N+\frac{1}{2})(x-t))}{\sin(\frac{1}{2}(x-t))}\,dt. $$ This has a $\delta$-function type of property as $N\rightarrow\infty$. For example, the integral where $f\equiv 1$ always has the value of $1$. The integral over any interval excluding $[x-\epsilon,x+\epsilon]$ tends to $0$ as $N\rightarrow\infty$. So the limit ends up plucking out the value of $f(x)$ if, for example, $f$ has a derivative at $x$.