Question

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?

Answer 1

See

1. 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).

2. 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.

3. 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).