  We introduce new discrete best approximation problems, formulated and solved
in the framework of tropical algebra, which deals with semirings and semifields
with idempotent addition. Given a set of samples, each consisting of the input
and output of an unknown function defined on an idempotent semifield, the
problem is to find a best approximation of the function, by tropical Puiseux
polynomial and rational functions. A new solution approach is proposed, which
involves the reduction of the problem of polynomial approximation to the best
approximate solution of a tropical linear vector equation with an unknown
vector on one side (a one-sided equation). We derive a best approximate
solution to the one-sided equation, and we evaluate the inherent approximation
error in a direct analytical form. Furthermore, we reduce the rational
approximation problem to the best approximate solution of an equation with
unknown vectors on both sides (a two-sided equation). A best approximate
solution to the two-sided equation is obtained in numerical form, by using an
iterative alternating algorithm. To illustrate the new technique developed, we
solve example approximation problems in terms of a real semifield, where
addition is defined as maximum and multiplication as arithmetic addition
(max-plus algebra), which corresponds to the best Chebyshev approximation by
piecewise linear functions.
