I proved that for $f \in \ell^1 (\mathbb Z)$ its Gelfand transform $\widehat{f}$ is a map $\widehat{f}: S^1 \to S^1$ defined by $$ \widehat{f}(z) = \sum_{n \in \mathbb Z}f(n) z^n$$
In Murphy's book it is stated that it is readily verified that the $f(n)$ are the Fourier coefficients of $\widehat{f}$. But what is there to show? Isn't it the definition of a fourier series of any function $g:S^1 \to S^1$ that $$ g(x) = \sum_{n \in \mathbb N} c_n e^{inx}$$
for some coefficients $c_n$? What would I have to show if I want to verify that $f(n)$ are the Fourier coefficients?