Let $f,g$ be periodic functions and $p>1$ such that,
$$ f\in L^p((-\pi,\pi)), g\in L^q((-\pi,\pi)) $$ Consider the function $$ h(x,t) = f(x+t)g(t) $$ prove that $h$ is measurable. Consider
$$ c_n(x) = \frac{1}{2\pi} \int_{-\pi}^{\pi} f(x+t)g(t)e^{-int}\;dt $$ Show that $\lim_{n\to\infty} = 0$ uniformly for $x\in \mathbb(R)$
Any suggestions for proving the limit is $0$ uniformly??
Hint: Let $\Lambda_n$ be defined on $L^1$ by the formula
$$\Lambda_n (h) = \int_{-\pi}^\pi h(t)e^{-int}\,dt.$$
Then $\{\Lambda_n\}$ is easily seen to be equicontinuous. By the Riemann-Lebesgue lemma, $\Lambda_n \to 0$ pointwise on $L^1.$ Apply Arzela-Ascoli to see $\Lambda_n \to 0$ uniformly on compact subsets of $L^1.$ Then show the functions $f(x+t)g(t), x\in [-\pi,\pi],$ form a compact subset of $L^1.$