Scale of x-axis for Fourier transform

Consider a function $f(t)$ and its Fourier transform $F(omega)$. The amplitude of the Fourier transform $F(omega)$ depends on the frequency $omega$ and thus also depends on the scale of the $t$-axis. For example, if $f_1(t)$ is a box function which is 1 between $[0,10^9]$ and $f_2(t)$ is a box function which is 1 between $[0,1]$, then the amplitude of $F_1(omega)$ is $10^9$ times larger than the amplitude of $F_2(omega)$.

However, numerically, this is not the case. If you have 100 points of the function $f_1(t)$ equally spaces between $[0,10times10^9]$ and also 100 points of the function $f_2(t)$ equally spaced between $[0,10]$, then numerically the amplitude of the Fourier transform of both functions will be the same (which is not correct). Numerically, you only consider the Fourier transform of the "y-values" $f(t_i)$, so the information about the $t$-axis scale is lost…

In this example of the Fourier transform of the rectangular function, you can solve this problem by multiplying the result with $10^9$ in the first case because the Fourier transform of the rectangular function is proportional with $1/omega$. Now, how to solve this problem for a general function $F(omega)$? For example, when the Fourier transform is proportional to $e^omega$ or proportional to $1/omega^2$ or …? Is there a numerical approach to take the scale of the "x-axis" into account?