Let $g :\partial\Omega\to\mathbb{R}$ where $\partial \Omega$ is unit circle. Let $h(\theta)$ be the polar form of $g(x,y)$. What is the condition on $g(x,y)$ to assure that $h(\theta)$ has fourier series representation?
This is actually from a partial differential equation, $$\nabla^2 u(x,y) = 0\qquad x^2 + y^2 < 1$$ with boundary condition $$u(x,y) = g(x,y)\qquad x^2 + y^2 = 1.$$
I can solve the problem, but the Prof didn't say anything about $g(x,y)$. This makes me unsure with steps I did in solving the PDE.
Thanks in advance.