Given a function $f(x)$, I want to find another function such that $<f, g> = 0$, in the space $R([0, pi])$ with the inner product $L^2$. The only way I've been able to do it is by trial and error. That is, I set up an integral an try on function and then evaluate the integral, to see if the result is 0.
But is there really no other, smarter way? I don't want a trivial solution ( $g = 0$).
If $\int f=0$, then any constant function is solution.
If $\int f \neq 0$, then for every function $h$, a solution is given by $$g(x)=h(x)- \frac{\int fh}{\int f}$$