Can you approximate a vector field? Say you have a physical simulation, there are "wind current" vectors stored in a 2d space.

So you know that the vectors near each other will likely be similar in direction.
Can we capitalize on the "similarity" across the vector field, and use it to write an approximation to the vector field?
So is there an alternative way to represent a vector field (something like a Fourier Transform for vector fields?)
 A: Your main point about vector fields is that vectors near each other are fairly similar.  Now, assuming we have well-behaved functions that do follow that behavior, one way of "capitalizing on the similarity of a vector field" is to simply "zoom out" by taking the average of all vectors within a certain area and getting the average vector in the center of the area.  This will yield a vector field with less vectors, which, because of our approximations, stand for a generalized behavior.
If, in the same scenario, we wished to emphasize the differences, then I'd suggest rather than taking an average vector for a rectangular area, you may want to divvy up your vector field such that there are regions defined by a 'central' vector and some range of tolerance from it.
However, what I think you are looking for is simply the divergence.  Given your vector field $\vec F(x, y)$, making a contour plot of $D(x, y) = \nabla \cdot \vec F(x, y)$ and then choosing values of $D$ to be the contours,  you can demonstrate how similar the direction and magnitude of a vector is by each curve's spacing.  This also causes a loss of information, which is what you were looking for.
