Assume I have a function $f(x)$ defined on an interval $[0,1]$. I'm wondering if it is possible (and is done in practice?..) to define a quantity $\widetilde{f}(x)$ (approximating $f(x)$ to certain extent?..) by $$ f(x) \stackrel{\text{Fourier transform}}{\longrightarrow} \hat{f}(k) \stackrel{``\text{Discrete Fourier transform''}}{\longrightarrow} \widetilde{f}(x) $$ Here $$ f(x) = \sum \limits_{k\in\mathbb{Z}} \operatorname{e}^{i2\pi xk} \hat{f}(k) \quad,\quad\text{where}\quad \hat{f}(k) = \int\limits_0^1 dy \operatorname{e}^{-i2\pi yk} f(y) $$ $\hat{f}(k)$ and $\widetilde{f}(x)$ are defined for integer values of $k$ and $x$.
I put quotation marks because the Discrete Fourier Transform requires a finite sum, while I'm interested in both options:
- $ \hat{f}(k) = \sum\limits_{k\in\mathbb{Z}} \tilde{f}(x) \operatorname{e}^{?} $
- $ \hat{f}(k) = \sum\limits_{k\in\text{some finite subset of }\mathbb{Z}} \tilde{f}(x) \operatorname{e}^{?} $
I would expect getting smth like a function defined on a discrete set of points of the interval $[0,1]$, especially in the second case.