# A question about Fejer kernel

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.