Dig a little further into the “bilinear filter” and “bicubic filter” that follow the box filter discussion. They are more interesting than the box filter because the contribution of a clipped polygon is not constant across the polygon fragment, unlike the box filter which is constant across each fragment. Integrating non-constant contribution is where Green’s Theorem comes in.
It’s also conceptually useful to understand the equivalence between box filtering with analytic computation and box filtering with multi-sample point sampling. It is the same mathematical convolution in both cases, but it expressed very very differently depending on how you sample & integrate.
If they called it a choice of basis or influence function, it would've been so much clearer.
I would think that conceptually that a basis function is different form a filter function because a basis function is usually about transforming a point in one space to some different space, and basis functions come in a set that’s the size of the dimensionality of the target space. Filters, even if you can think of the function as a sort of basis, aren’t meant for changing spaces or encoding & decoding against a different basis than the signal. Filters transform the signal but keep it in the same space it started from, and the filter is singular and might lose data.
May be better if I just link to what others say about filters than me trying to blabber on https://en.wikipedia.org/wiki/Filter_(signal_processing)
