# Proof of Fejer's lemma

How does ont prove Fejer's lemma:

If $f \in L^1(\mathbb{T})$ and $g\in L^\infty(\mathbb{T})$, then $\lim_{n \rightarrow \infty} \int f(t) g(nt) \, dt = \hat{f}(0)\hat{g}(0).$

• Approximate f i the $L^1$ by polynomials – usere5225321 Jan 22 '18 at 14:57
• I think that the following path could bring you to a proof. Fix an arbitrary trigonometric polynomial $f$ and an arbitrary characteristic function $g$ of an interval $(a,b)$. Prove the conclusion using brute force in this case. Then by the $\pi-\lambda$ theorem (or something like that), extend the result for an arbitrary characteristic function $g$ of a Borel-measurable set. Then, by linearity and bounded convergence, extend the result for an arbitrary $g\in L^\infty$ function. Finally, by the density of trigonometric polynomials in $L^1$, extend the result to get the conclusion – Bob Aug 21 '18 at 10:15

Show that it's enough to prove it assuming $f$ is a trigonomtric polynomial. Figure out the Fourier coefficients of $g(nt)$ in terms of $\hat g(k)$.
• C.Ullrich I have seen that continuous functions on thr circle can be uniformly approximated by trigonometric polynomials. So I assume it is enough to show it for $f$ trigonometric. Also, $\hat{g}(nt)=\hat{g} (t/n)$ if t|n and 0 otherwise. Now im stuck – usere5225321 Jan 22 '18 at 18:27