I was trying to separate the periodicity aspect of the unit circle in the Fourier transform, by creating some patches:
$$f(x) = \sum_{n=-\infty}^{\infty} \mathbb{1}_{[0,\lambda]} (x-n\lambda) f(x), $$
where $\mathbb{1}_{[0,\lambda]}(x)$ is the indicator function. Now let's define the patches as,
$$f^n(\lambda,x) = \mathbb{1}_{[0,1]}(x) f(\lambda x + \lambda n),$$
which limits the domain to $[0,1]$. Lets then define:
$$\hat{f}(k)=\sum_n \int_0^{1} w(x) f^n \left(2\pi k^{-1},x\right) dx.$$
If $w(x)=e^{2\pi i x}$, is $\hat{f}(k)$ the same as the usual Fourier transform?