Finite-dimensional Approximations of Push-forwards on Locally Analytic Functionals and Truncation of Least-Squares Polynomials

Core Concepts

This paper introduces a theoretical framework for investigating analytic maps from finite discrete data, elucidating the mathematical machinery underlying polynomial approximation with least-squares in multivariate situations.

Abstract

The key insights and highlights of the content are:

The paper proposes an approach that does not directly approximate the analytic map itself, but instead approximates the push-forward on the space of locally analytic functionals induced by the analytic map. This enables a theoretical handling of approximation with least-squares using multivariate polynomials in more general situations.

The paper establishes a methodology for appropriate finite-dimensional approximation of the push-forward from finite discrete data, through the theory of the Fourier–Borel transform and the Fock space. It proves a rigorous convergence result with a convergence rate.

As an application, the paper proves that it is not the least-squares polynomial, but the polynomial obtained by truncating its higher-degree terms, that approximates analytic functions and allows for approximation beyond the support of the data distribution.

The paper shows that the finite-dimensional approximation of the push-forward enables the application of linear algebraic operations. Utilizing this, it proves the convergence of a method for approximating an analytic vector field from finite data of the flow map of an ordinary differential equation.

The paper provides a general theoretical framework for estimating analytic maps from finite discrete data, shedding light on the underlying machinery of polynomial approximation with least-squares in multivariate situations.

Stats

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Quotes

"Our approach is to consider the push-forward on the space of locally analytic functionals, instead of directly handling the analytic map itself."
"One advantage of our theory is that it enables us to apply linear algebraic operations to the finite-dimensional approximation of the push-forward."

Key Insights Distilled From

Finite-dimensional approximations of push-forwards on locally analytic functionals and truncation of least-squares polynomials

by Isao Ishikaw... at arxiv.org 04-17-2024

https://arxiv.org/pdf/2404.10769.pdf

Deeper Inquiries

How can the proposed framework be extended to handle non-analytic maps or functions

To extend the proposed framework to handle non-analytic maps or functions, we can consider generalizing the concept of the push-forward on locally analytic functionals to a broader class of functionals. This could involve incorporating distributional or generalized function spaces to accommodate non-analytic functions. By working with function spaces that allow for a wider range of functions, such as Sobolev spaces or spaces of tempered distributions, we can adapt the methodology to handle non-analytic maps. Additionally, techniques from functional analysis and harmonic analysis can be employed to extend the framework to non-analytic settings, allowing for a more comprehensive approach to approximation and modeling.

What are the potential limitations or drawbacks of the truncation-based approach for polynomial approximation compared to other techniques

While the truncation-based approach for polynomial approximation offers several advantages, such as simplicity and computational efficiency, there are potential limitations and drawbacks to consider. One limitation is the loss of information when higher-degree terms are truncated, which can lead to inaccuracies in the approximation, especially for functions with complex or rapidly changing behavior. Additionally, the convergence of the truncated polynomial approximation may be slower or less accurate compared to other techniques, such as interpolation or spline methods. The choice of the truncation level can also impact the quality of the approximation, and determining the optimal degree of truncation can be challenging, especially for functions with unknown characteristics. Overall, while truncation-based approaches are useful for certain applications, they may not always provide the most precise or reliable results for all types of functions.

How can the insights from this work be applied to improve data-driven analysis and modeling in other domains beyond dynamical systems and ordinary differential equations

The insights from this work can be applied to improve data-driven analysis and modeling in various domains beyond dynamical systems and ordinary differential equations. For example, in machine learning and data science, the concept of finite-dimensional approximations and truncation techniques can be utilized for feature engineering, dimensionality reduction, and model simplification. By leveraging the idea of approximating complex functions with finite-dimensional spaces, researchers and practitioners can develop more efficient algorithms, reduce computational complexity, and enhance the interpretability of models. Additionally, the convergence results and methodologies presented in the framework can be adapted to optimize model performance, enhance predictive capabilities, and facilitate the analysis of large-scale and high-dimensional data in diverse fields such as image processing, signal processing, and optimization.