Let $f\in L^{\infty}(\mathbb{T}^n)$ and $g\in L^2(\mathbb{T}^n)$, then their product $fg$ is also in $L^2(\mathbb{T}^n)$.
Now, there are two ways in which the Fourier series of $fg$ could be written. One by considering it as a function $fg$ in $L^2(\mathbb{T}^n)$, and another, by defining the coefficients of its Fourier series, as the convolution $\sum_k\hat{f}(n-k)\hat{g}(k)$.
- Are these Fourier series equal? This seems to be true by some kind of uniqueness of Fourier coefficients.
- Does it follow that this Fourier series converges in norm to the function?