This is a problem I found in Schaum's Outlines: Linear Algebra, and I was wondering if someone knew how to solve it. I began using integration by parts, but that approach did not lead to any conclusions.
Let V be the space of all infinitely-differentiable functions on R which are periodic of period h>0 [i.e., f(x+h) = f(x) for all x in R]. Define an inner product on V by $$\langle f,g\rangle =\int_{-h}^hf(x)g(x)dx$$
Let $\alpha(f)=f'$. Find $\alpha^*$.
I know that the adjoint implies the relationship $\langle\alpha(f),g\rangle= \langle f,\alpha^*(g)\rangle$ .
Thank you.
Best Answer
We have $$\langle \alpha(f),g\rangle=\int_{-h}^hf'(t)g(t)dt=\left[f(t)g(t)\right]_{-h}^h-\int_{-h}^hf(t)g'(t)dt.$$ As $f\cdot g$ is periodic of period $h$, $\left[f(t)g(t)\right]_{-h}^h=0$, so $\alpha^*(f)=-\alpha(f)$.