No solution please.
I want to show that the following limit exists $$ \lim_{T\to \infty}\frac{1}{2T}\int_{-T}^{T}f(x)\,\overline{g(x)}\,dx $$
where $f$ and $g$ are finite linear combinations of complex exponentials like
$f(x)=\sum_{k=1}^{N}a_{k}\mathrm{e}^{i\lambda_{k}x}$, where $x$ and $\lambda$ are real numbers.
Why calculating it isn't enough?
Thanks
Clearly, the product $\,f\,\overline{g}\,$ is a linear combination of functions of the form $\mathrm{e}^{i\lambda x}$, where $\lambda$ is real. Hence, it suffices to show that, for every $\lambda\in\mathbb R$ the limit $$ \lim_{T\to\infty}\frac{1}{2T}\int_{-T}^T \mathrm{e}^{i\lambda x}\,dx $$ exists. If $\lambda=0$, then the above limit equals to $1$. If $\lambda\ne 0$, then $$ \frac{1}{2T}\int_{-T}^T \mathrm{e}^{i\lambda x}\,dx=\frac{1}{2T}\frac{1}{i\lambda} \big(\mathrm{e}^{i\lambda T}-\mathrm{e}^{-i\lambda T}\big)=\frac{\sin(\lambda T)}{T}\to 0. $$