I am studying Gelfand transform and Wiener's theorem from the lecture notes given by our instructor. Here I came across the following definition about Fourier coefficients of a continuous function defined on $\mathbb S^1.$ Here is the definition $:$
Definition $:$ Let $f \in C \left (\mathbb S^1 \right ).$ Then one defines the $n^{\text {th}}$ Fourier coefficient of $f$ by $\widehat {f} (n) = \int_{\mathbb S^1} f(z)\ z^{-n}\ dz,$ where $n \in \mathbb Z$ and the integration is with respect to the Haar measure on the group $\mathbb S^1.$
Just after the definition a result has been mentioned as a remark which states the following $:$
If two continuous functions defined on $\mathbb S^1$ have the same Fourier coefficients of all order then they are the same.
But I find it difficult to prove this fact. Could anyone kindly help me in showing this?
Thanks for investing your valuable time in reading my question.