Let $D$ be the unit disk, i.e. $D=\{(x,y): x^2+y^2\le1\}$. Let $f\colon D\to \mathbb R$ be a function.
I associate to $f$ a new function $F\colon [0,1]\times [0,2\pi]\to \mathbb R$ such that
- $F(r,\theta)=f(r\cos\theta,r\sin\theta)$ for all $(r,\theta)\in [0,1]\times [0,2\pi]$;
- $F(0,\theta)$ is indepdente from $\theta\in [0,2\pi]$
- $F(r,0)=F(r,2\pi)$ for all $r\in [0,1]$
If $f\in L^2(D)$ then $F(r,\cdot)\in L^2([0,2\pi])$ because $$ \int_D |f(x,y)|^2 dxdy=\int_0^1 rdr \int_0^{2\pi} |F(r,\theta)|^2d\theta\,.$$ As $F(\cdot,0)=F(\cdot,2\pi)$, I can expand $F(\cdot,\theta)$ in Fourier series. So I obtain $$ F(r,\theta)=\sum_{n\in \mathbb Z} f_n(r)e^{in\theta}$$ for some $(f_n(r))_{n\in\mathbb Z}\in \ell^2(\mathbb Z)$.
Is this expansion valid?
Thanks in advance.
This has been essentially answered in comments to the main question. Summarizing, the expansion is valid. For more information on this problem and its higher-dimensional generalizations, see "Introduction to Fourier analysis on Euclidean spaces" of Stein and Weiss, chapter IV: "Symmetry properties of the Fourier transform".