Let $f(x)=g(x)$ for all $x$ in $[-\pi,\pi]$ except when $x=0$. Show that $f(x)$ and $g(x)$ have the same Fourier Coefficients?

Question

Question:

Let $f(x)=g(x)$ for all $x$ in $[-\pi,\pi]$ except when $x=0$. Show that $f(x)$ and $g(x)$ have the same Fourier Coefficients?

My Attempt:

So I've tried to calculate the fourier coefficients but i can only show that $a_0^f=a_0^g$, not $a_n^f=a_n^g$ and $b_0^f=b_0^g$. I also tried letting $h(x)=f(x)-g(x)=0$ but I don't think that works since $h(0)=f(0)-g(0)\neq0$.

Thanks for the help.

Answer 1

By definition, the Fourier coefficients of $f$ are given by $$ \hat{f}(n)=\frac{1}{\pi}\int_{-\pi}^{\pi}f(x)e^{-inx}dx $$

So changing one point of $f$ does not change the coefficients.

On the other hand, if $f$ is continuous on $[-\pi,\pi]$ and $\hat{f}(n)=0$ for all $n$, then $f\equiv 0$.