  This paper introduces a theoretical framework for investigating analytic maps
from finite discrete data, elucidating mathematical machinery underlying the
polynomial approximation with least-squares in multivariate situations. Our
approach is to consider the push-forward on the space of locally analytic
functionals, instead of directly handling the analytic map itself. We establish
a methodology enabling appropriate finite-dimensional approximation of the
push-forward from finite discrete data, through the theory of the
Fourier--Borel transform and the Fock space. Moreover, we prove a rigorous
convergence result with a convergence rate. As an application, we prove that it
is not the least-squares polynomial, but the polynomial obtained by truncating
its higher-degree terms, that approximates analytic functions and further
allows for approximation beyond the support of the data distribution. One
advantage of our theory is that it enables us to apply linear algebraic
operations to the finite-dimensional approximation of the push-forward.
Utilizing this, we prove the convergence of a method for approximating an
analytic vector field from finite data of the flow map of an ordinary
differential equation.
