Fourier coefficients of $f^2$ for $f\in L_4([0,1])$

Question

I'm trying to do the following problem:

For a function $f\in L_1([0,1]) $ recall that $$ \hat{f}(k)=\int_0^1 e^{-2\pi i kx} f(x) dx.$$

Show that for $f\in L_4([0,1])$ the sequence $$ c_n =\Sigma_{k+j=n} \hat{f}(k)\hat{f}(j)$$ converges to $0$ for $n\to \pm \infty$.

I'm thinking of relating $c_n$ with the Fourier coefficient of $f^2$ by usingFourier transform of convolution: $$\int _{0}^{1} f^2(x) e^{-2\pi ikx}= \frac{1}{2\pi} \int_0^1 \hat{f}(x) \hat{f} (2\pi k-x)dx$$ which converges to $0$ as $k\to \infty$. But I don't know if one can compare this coefficient with $c_n$. Can anyone give me some hints on how to solve this problem?

Answer 1

HINT: what do you know about the convergence of $\sum \hat{f}(k)e^{2\pi i k x}$ to $f$?

HINT2: For $g \in L^a[0,1]$, with $1\leq b < a \leq \infty$ we have $g\in L^b[0,1]$ as a consequence of Jensen inequality or Holder inequality.