Interpretation of $\tilde{f}(\mathbf{0})$

Question

Given a function in real space $f(\mathbf{r})$, what is the interpretation of the of the value $\tilde{f}{(\mathbf{0})}$?

As an example, take the Fourier transform of \begin{equation} V(\mathbf{r}) = \begin{cases} V_0 & r<r_0 \\ 0 & r>r_0 \end{cases} \end{equation} to be \begin{equation} \tilde{V}(\mathbf{k}) = V_0 \dfrac{4\pi r_0^2}{k} j_1 (kr_0) \end{equation}

where $k=|\mathbf{k}|$ and $j_1$ is the first-order spherical Bessel function and show that the 'correct' value is obtained in the limit $k\rightarrow 0$.

Answer 1

I assume the tilde references to the Fourier transform of $f(r)$.

Each value $\tilde{f}(k)$ tells us about the amplitude of change with period $k$. If $k=0$ this "change with period $k$" isn't a change at all - it's a constant. In other words $\tilde{f}(0)$ is the offset of $f$. In particular, if $V(x) = 1$ then the transform will give us a delta peak at $k = 0$ (I am doing it in 1D for ease).

If you compute the Fourier Transform of the potential in the limit of $k \to 0$ you will come up at $\tfrac{4}{3} \pi r_0^3V_0$ and that is the total offset of this function.