Integration of periodic functions I have a question at hand (which may be easy to some) ,but unfortunately I don't know how to even begin with. Could someone help me?
If $f$ and $g$ are continuous, $2\pi$ periodic functions then prove that
$$\lim_{n\to \infty} {1\over 2\pi} \int_{-\pi}^\pi f(t)g(nt)dt=\left({1\over2\pi}\int_{-\pi}^\pi f(t)dt\right)\left({1\over 2\pi}\int_{-\pi}^\pi g(t)dt \right)$$
Is this even remotely connected to Cauchy-Schwarz?
 A: Interesting identity.
To grasp it, consider that the integrals on the right are just the average values of the two functions, i.e:
$$
{1 \over {2\pi }}\int_{ - \pi }^\pi  {f(t)\,dt}  = {\rm avg}\left( {f(t)} \right) = \overline f 
$$
so that we can write: $
f(t) = \overline f \; + \tilde f(t)\quad  \to \quad \int_{ - \pi }^\pi  {\tilde f(t)\,dt}  = 0$
and : $\quad \quad \quad \quad \quad \quad g(t) = \overline g  + \tilde g(t) = \overline g  + \tilde g(nt)\quad \left| {\;1 \le {\rm integer }\ n} \right.$
Therefore:
$$
\eqalign{
  & {1 \over {2\pi }}\int_{ - \pi }^\pi  {f(t)\;g(n\,t)\,dt}  = {1 \over {2\pi }}\int_{ - \pi }^\pi  {\left( {\overline f \; + \tilde f(t)} \right)\;\left( {\overline g \; + \tilde g(n\,t)} \right)\,dt}  =   \cr 
  &  = {1 \over {2\pi }}\int_{ - \pi }^\pi  {\overline f \;\overline g \,dt}  + {1 \over {2\pi }}\int_{ - \pi }^\pi  {\overline f \;\tilde g(n\,t)\,dt}  + {1 \over {2\pi }}\int_{ - \pi }^\pi  {\tilde f(t)\;\overline g \,dt}  + {1 \over {2\pi }}\int_{ - \pi }^\pi  {\tilde f(t)\;\tilde g(n\,t)\,dt}  =   \cr 
  &  = \overline f \;\overline g \;\,{1 \over {2\pi }}\int_{ - \pi }^\pi  {dt}  + \overline f {1 \over {2\pi }}\int_{ - \pi }^\pi  {\;\tilde g(n\,t)\,dt}  + \;\overline g {1 \over {2\pi }}\int_{ - \pi }^\pi  {\tilde f(t)\,dt}  + {1 \over {2\pi }}\int_{ - \pi }^\pi  {\tilde f(t)\;\tilde g(n\,t)\,dt}  =   \cr 
  &  = \overline f \;\overline g \;\,{{2\pi } \over {2\pi }} + \overline f \;0 + \;\overline g \;0 + {1 \over {2\pi }}\int_{ - \pi }^\pi  {\tilde f(t)\;\tilde g(n\,t)\,dt}  =   \cr 
  &  = \overline f \;\overline g \;\, + {1 \over {2\pi }}\int_{ - \pi }^\pi  {\tilde f(t)\;\tilde g(n\,t)\,dt}  \cr} 
$$
and the last integral is the average value of the "high-frequency" carrier $
{\tilde g(nt)}$ , amplitude modulated by ${\tilde f(t)}$.
