As someone who mostly does symbolic computation, I've always been puzzled by the fascination mathematicians seem to have with L^{p}(R) (for p<∞)? To be more precise, there are **no** non-trivial polynomials in that space and, to me, polynomials are not only the simplest functions, they are the building blocks of most everything which can be (easily) manipulated algorithmically. And *restricting to a compact support* is really a non-answer, since one of the great things about polynomials is that they are global, analytic functions.

To ask a more precise question: are there some spaces of (total, real-valued) functions which are both nice from a functional analysis point of view, and contain all the polynomials?