I'm reading something about the Fejer Kernel on the space $\mathbb{T}$.
Now yesterday I find this affirmation:
If $f\in\mathcal{L^1(\mathbb{T})}$ and $g\in\mathcal{L^\infty ({\mathbb{T})}}$ than: $lim_{n \rightarrow \infty} \int_{-\pi}^{\pi}f(t)g(nt)\frac{dt}{2\pi}=\hat{f}(0)\hat{g}(0)$.
Now there is not a proof about this. The problem is that I don't understand how this can be prove. Someone can help me?
This follows from a sequence of well known measure-theoretic results. If $g$ is a trigonometric polynomial the result is Riemann-Lebesgue. So the strategy is to approximate $g$ by such in an appropriate norm but unfortunately, we cannot do it directly in $L^{\infty}$.
However, if $f$ is in $L^p, p>1$ then $g$ is in the dual space $L^q$ (bounded functions on the circle are in all $L^p$) and since now $q$ finite as $p>1$, trigonometric polynomials are dense in the $q$ norm and we can approximate $g$ by them and the result holds.
In general we then first approximate $f$ in the $L^1$ norm by continuous functions and it immediately follows that we just need to prove the result for such as $g(nx)$ is uniformly bounded by $||g||$ and we cann switch limits etc, but continuos functions are in any $L^p$ as they are bounded so the general result holds by all the above.