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.

这篇论文介绍了一种用于研究有限离散数据的解析映射的理论框架，阐明了多元情境中最小二乘多项式逼近的数学机理。通过考虑局部解析泛函空间的推前作用，而非直接处理解析映射本身，我们确立了一种从离散数据中适当进行推前的有限维逼近方法，通过 Fourier- Borel 变换和 Fock 空间的理论。此外，我们证明了一个带有收敛速率的严格收敛结果。作为应用，我们证明了逼近解析函数的并进一步实现超过数据分布支撑的多项式不是最小二乘多项式，而是通过截断其高阶项得到的多项式。我们理论的一个优势是它使我们能够对推前的有限维逼近应用线性代数运算。利用这一点，我们证明了从普通微分方程的流图的有限数据逼近解析向量场的方法的收敛性。