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?
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 ${\tilde f(t)}$.