I have a Lemma, which is central in Fourier analysis of piece-wise function that I need to prove. I have proposed a proof, however it seems to be a little "shallow".
The Lemma is:
The indicator functions $\chi_{A_j}(t)$(where ${A_j}$ is some interval $[t_i,t_{i+1}]$) and the piece-wise functions $f_n(t)=\sum_{j=1}^n\eta_j\chi_{A_j}(t)$ have the same Fourier coefficients on a uniform grid on the interval $[-a, a]$ for any $a$.
and my proposed proof:
Take the $\alpha_k$ Fourier coefficient of an indicator function: \begin{equation} \beta_k=\frac{1}{a}\int_{-a}^{a}\chi_{A_j}\sin k\omega t \text{d}t=\frac{1}{k \omega a}\chi_{A_j}\sin k \omega t \bigg|_{-a}^a=\frac{2}{k\pi}\chi_{A_j}\sin (k\pi) \end{equation} then we obtain a function independent of $a$. Hence, the Fourier coefficients are the same for any $a$, since they are independent of $a$.
But it is too shallow. It needs more substance. I have no idea what. Does anyone have a suggestion?
Thanks