I've stumbled across the following problem when studying about Fourier Analysis and are very interested in the solution:
For $f\in L^1(\mathbb{T})$ and $P$ a trigonometric polynomial of degree $N>0$ on $\mathbb{T}$, find calculate the Fourier coefficients of the product $fP$ in terms of those of $f$ and $P$.
Now, what I've tried is computing $\widehat{fP}$ by the following:
\begin{equation} \widehat{fP}=\frac{1}{2\pi}\int_{-\pi}^{\pi}f(x)P(x)e^{-inx}dx \end{equation}
From here, I tried substituting $P(x)=\sum_{m=-N}^{N}c_me^{im x}$ and by changing integral and sum I got
\begin{equation} \sum_{m=-N}^{N}c_m\widehat{f}(n-m) \end{equation}
Now, I'm already doubting whether my answer is correct or not, but if it were correct, how does this representation represent the Fourier coefficients of $P$ at all?
I look forward to any help! Thanks in advance.