Find functions in $L^2([0,2\pi])$ such that are orthogonal to $\sin(nx)$ and $\cos(nx)$

Question

I have to find functions $u \in L^2([0,2\pi])$ such that these two hold: $$ \int_0^{2\pi} u(x)\sin(nx) dx = 0 \qquad \int_0^{2\pi} u(x)\cos(nx) dx = 0$$ for any $n = 1,2,3,...$

I tried to argue considering them like the scalar product in $L^2$ of $u(x)$ with $\sin(nx)$ and $\cos(nx)$ respectively and using the fact that the subspace of even functions is the orthogonal complement of the subspace of odd functions. Is that right? Or how can I do in other ways?

Thank so much!

Answer 1

Write down the Fourier series for $u(x)$! Those two integrals you have written down are, apart from a scaling factor, the Fourier coefficients $a_k$ and $b_k$ for $u$. $$ u(x)\sim c_0+\sum_{k=1}^\infty \left(a_k\cos{kx}+b_k\sin{kx}\right)= c_0+\sum_{k=1}^\infty \left(0\cos{kx}+0\sin{kx}\right)=c_0. $$ This means that $u(x)$ must be constant.