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.

翻译：本文提出了一个理论框架，用于从有限离散数据研究解析映射，阐明了多元情形下最小二乘多项式逼近的数学机制。我们不再直接处理解析映射本身，而是考虑局部解析泛函空间上的推前映射。通过傅里叶-博雷尔变换与福克空间理论，我们建立了一种方法论，能够从有限离散数据对推前映射进行恰当的有限维逼近。此外，我们证明了带有收敛速度的严格收敛性结果。作为应用，我们证明了逼近解析函数的并非最小二乘多项式，而是通过截断其高次项得到的多项式，并且该多项式还能在数据分布支撑集之外实现逼近。本理论的一个优势在于，它使我们能够对推前映射的有限维逼近应用线性代数运算。利用这一特性，我们证明了通过常微分方程流映射的有限数据逼近解析向量场的方法的收敛性。