# Generalized Fourier series

This is my first post here, so I hope I'm not doing sth wrong. I have to prove the following statement:

For $f \in C^1(\mathbb{R^2})$ define $f_n(x,y) = \int_{\mathbb{T}} f(k(\theta)(x,y))e^{-2\pi i n\theta}\mathrm{d}\theta$

where $k(\theta) = \begin{pmatrix} \cos \theta & -\sin\theta \\ \sin\theta & \cos \theta \\ \end{pmatrix}$ and $\mathbb{T} = \mathbb{R}/\mathbb{Z}$.

Then we have $f(x,y) = \sum_{n\in\mathbb{Z}} f_n(x,y)$ and even assuming $f \in L^2(\mathbb{R^2})$ one still has convergence in $L^2$.

My problem is, that I lack a reasonable point to start so I would like some hints or suggestions, since this strikes me as an interesting statement.

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Remark : I think you mean $k(2\pi\theta)$ (instead of $k(\theta)$) in the integral. Hint : for a given $(x,y) \in \mathbb{R}^2$, the function $g : \theta \mapsto f(k(2\pi\theta) (x,y))$ is $C^1$ (or $L^2$) and $1$-periodic. –  Joel Cohen Nov 1 '11 at 21:10