I'm trying to understand the underlying logic in the Radial Basis Function interpolation.
I understood that we estimate the value of the underlying function in any unknown point as
$ y(\vec{x}) = \sum \lambda_j \phi(r_j) $
where $r_j$ is the distance between the query point and $\vec{x}$.
In my understanding, we want the value of $y(\vec{x})$ to be "more similar" to the value of the underlying function in the points closer to $\vec{x}$. For this reason it makes sense to me to use basis functions as the inverse multiquadric or the gaussian that tends to zero as $r$ tends to infinity so that the "influence" of a point very far from my query point tends to zero.
However, reading the literature, I noticed that the most common and "original" basis function is the multiquadric one, defined as
$ \phi(r) = \sqrt{\left(r/\varepsilon\right)^2 + 1} $
that tends to $\infty$ as $r \rightarrow \infty $.
Could you please help me understanding why some basis function tends to zero, while other tends to infinity as $r \rightarrow \infty $, and still are both equally acceptable?