Minimize the term $\int_{-\pi}^{\pi}|x-a-be^{3ix}-c^{5ix}|^2dx$ for $a,b,c \in \mathbb{R}$
Thoughts-
So it's a rather common problem, i know that if $ \{1, e^{3ix}, e^{5ix} \} $ span a vector space, then the minimum of the term is the minimum of $ || x -g|| $ for $g \in span\{1, e^{3ix}, e^{5ix} \}$. therefore, the minimum will be the projection of $x$ onto $span\{1, e^{3ix}, e^{5ix} \}$. That is -
$a = \dfrac{<x,1>}{||1||^2} , b = \dfrac{<x,e^{3ix}>}{||e^{3ix}||^2} , c = \dfrac{<x,e^{5ix}>}{||e^{5ix}||^2}$
Is it right? what do you think?