# Is there a tropical analogue of a reproducing kernel Hilbert space?

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 algebra. Both of the properties I mentioned above have max-plus analogues (see this 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?

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