In classical functional analysis, one can construct a reproducing kernel Hilbert space by starting with a positive definite kernel, say
$K: [0,1]\times [0,1] \rightarrow \mathbb{R}$.
One then creates linear combinations of the form
$f(x) = \sum^n a_i k(x_i,x)$, together with an inner product

$\langle f,g \rangle = \sum \sum a_i b_j k(x_i,x_j),$

and completes the space as usual.

As far as I can see, the 'essential' properties of a reproducing kernel Hilbert space are
the ability to represent linear operators as integral kernels, and the Riesz representation theorem.

I'm just wondering if there is a similar construction to the one outlined above in the framework of the [max-plus][1] algebra. Both of the properties I mentioned above have max-plus analogues (see [this][2] introduction). If such a construction exists, how far is it possibe to take it? Is there an idempotent version of Mercer's theorem, for example?


  [1]: https://en.wikipedia.org/wiki/Max-plus_algebra
  [2]: https://arxiv.org/pdf/math/0507014v1.pdf