This is exercise 1.5 from Analysis of Boolean Functions by Ryan O'Donnell.
Suppose $f: \{-1, 1\}^n \to \{-1, 1\}$. We want to show that at most one Fourier coefficient has magnitude exceeding $1/2$.
Fourier coefficient is defined as $\hat{f}(S) = \frac{1}{2^n} \sum_{x \in \{-1, 1\}^n} f(x) (\prod_{i \in S} x_i)$.
I do not think Parseval's identity will help here since we can have a non-boolean $L^2$ norm $1$ function that has more than one Fourier coefficient exceeding $1/2$.
Any hint on this? Thank you.