TITLE: Is there a tropical analogue of a reproducing kernel Hilbert space?
QUESTION [6 upvotes]: 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?

REPLY [8 votes]: See

G.L. Litvinov, V.P. Maslov and G.B. Shpiz. Idempotent functional analysis.
An algebraic approach // Mathematical Notes, v. 69, # 5, 2001, p. 696-729. 
E-print math.FA/0009128  (http://ArXiv.org).
G.L. Litvinov and G.B. Shpiz. Kernel  theorems  and nuclearity in 
idempotent  mathematics.  An  algebraic  approach,  Journal of 
Mathematical Sciences, v. 141, #4, 2007, p. 1417-1428. See also 
E-print math.FA/0609033 (http://arXiv.org), 2006.
G.L. Litvinov. Tropical mathematics, idempotent analysis, classical
mechanics and geometry. - in: Spectral Theory and Geometric Analysis M.Braverman et al., 
Eds.,  AMS Contemporary Mathematics, vol. 535, 2011, p. 159-186. See also E-print 
arXiv: arXiv: 1005.1247 (http://arXiv.org).