Show that for $f\in L_{4}([0,1])$ the sequence $$c_n = \sum_{k+j = n} \hat{f}(k) \hat{f}(j)$$ converges to $0$ for $n\to \pm \infty$.
Attempt: We can write $c_n$ as $$c_n = \sum_{j\in \mathbb{Z}} \hat{f}(n-j) \hat{f}(j) = \hat{f^2}(n).$$ Do we have that Fourier coefficients decay to $0$ here? Otherwise it seems we should use Bessel's inequality or Parseval's identity: $$\sum_{n=1}^{\infty} |c_n|^2\leq \|f^2\|_{2}<\infty$$ because $f\in L_{4}([0,1]) \Longrightarrow f^2\in L_{2}([0,1])$.