Let $f(t)$ be a function defined on the finite interval $[t_1,\, t_2]$. Is the Fourier transform of such a function uniquely defined? In the sense that there exists only one function $\hat{f}(\omega)$ giving the Fourier coefficients that sum up to $f(t)$ in the interval $[t_1,\,t_2]$?
I am aware that one can find the Fourier transform of a function $g_1(t)$ defined on $[-\infty, \,+\infty]$ as
$$g_1(t) =
\begin{cases}
f(t) & \text{for}\;t \in [t_1, t_2]\\
0 & \text{for}\;t \lt t_1 \;\text{and}\;t \gt t_2
\end{cases}
$$
The Fourier transform of this function would yield the coefficients that sum up to $f(t)$ on the interval $[t_1,\,t_2]$, I think. However, the same could be said about functions $g_2(t)$ and $g_3(t)$ defined via
$$g_2(t) =
\begin{cases}
f(t) & \text{for}\;t \in [t_1, t_2]\\
4 & \text{for}\;t \lt t_1 \;\text{and}\;t \gt t_2
\end{cases}
$$
$$g_3(t) =
\begin{cases}
f(t) & \text{for}\;t \in [t_1, t_2]\\
t & \text{for}\;t \lt t_1 \;\text{and}\;t \gt t_2
\end{cases}
$$
or any other function that matches the definition of $f(t)$ on $[t_1,\,t_2]$ and somehow continues this to be defined on $[-\infty,\,+\infty]$. In particular, by continuing $f(t)$ periodically on $[-\infty,\,+\infty]$, one would get a Fourier series that would also be a valid representation of $f(t)$ on $[t_1,\,t_2]$. Or is this reasoning incorrect?
If not, in what way does the concept of a Fourier transform apply to functions that are only defined on a finite interval?
Best Answer
In mathematics, there are many Fourier transforms. For every locally compact abelian group, there is a corresponding Fourier transform. Two of these Fourier transforms can be applied to a function $$f:[t_1,t_2]\rightarrow\mathbb{C}.$$ 1) You can apply the Fourier transform $\mathscr{F}_{\mathbb{T}}$, where $\mathbb{T}:=\mathbb{R}/(t_2-t_1)\mathbb{Z}$ is a torus group. An application of $\mathscr{F}_{\mathbb{T}}$ corresponds to the action of extending $f$ periodically and then expanding it into a Fourier series. If you choose $\mathscr{F}_{\mathbb{T}}$ as your Fourier transform, $\mathscr{F}_{\mathbb{T}}(f)$ will be unique.
2) You can also choose to apply the Fourier transform $\mathscr{F}_{\mathbb{R}}$, but then you have to extend $f$ from $[t_1,t_2]$ to $\mathbb{R}$ beforehand. As you pointed out yourself, there is no unique way of doing this. Consequently, there is no unique way to define $\mathscr{F}_{\mathbb{R}}(f)$.
Addition (Discrete Fourier Transform):
If you consider the finite abelian group $\mathbb{Z}_N:=\mathbb{Z}/N\mathbb{Z}$, the corresponding Fourier transform $\mathscr{F}_{\mathbb{Z}_N}$ is called distrete Fourier Transform. If you only have $N$ sampling points $f(x_0),f(x_1),\ldots,f(x_{N-1})$, you may think of it as a function on $\mathbb{Z}_N$ and consequently apply the Fourier transform $\mathscr{F}_{\mathbb{Z}_N}$ to it.
I will not define the Fourier transforms $\mathscr{F}_{\mathbb{T}}$, $\mathscr{F}_{\mathbb{R}}$, $\mathscr{F}_{\mathbb{Z}_N}$ here (you can easily find good sources elsewhere), but simply emphasize that there is more than one Fourier transform, which may resolve some of the perplexity behind the question in the OP.