Convex basis of functions

Question

I'm looking for a set of convex functions which is forms a basis for $C^1(\mathbb{R})$?

Most of the basises I know are polynomials or Fourier basis but I was wondering if there was a basis of convex functions.

Answer 1

A basis of convex functions is a tricky concept. Convexity is not closed under operations over vector spaces: a linear superposition of convex functions does not have to be convex (in fact, it most likely isn't). So... the entire formalism of vector spaces (Hilbert spaces actually) goes through the window and the concept of a basis doesn't make much sense.

The problem is, that when you develop something over a basis, you may get negative coefficients. The question is actually exactly the same as asking for a basis of positive functions (you get the basis of convex functions by integrating the basis of positive functions twice).

I don't know exactly what you want to achieve. It's of course possible to have a decomposition of a positive-definite function into a sum of positive definite functions, it's just not a vector-space based decomposition, so there is no orthogonality, superposition principle, guarantee of completeness, and so on. Maybe one of the other readers knows an example of this. In discrete world, you'd just get a linear programming question, but I don't know how to deal with the continuous version.