Whitney extensions on symmetric spaces (2407.21420v2)

Abstract: In 1934, H. Whitney introduced the problem of extending a function on a set of points in $\mathbb{R}^n$ to an analytic function on the ambient space. In this article we prove Whitney type extension theorems for data on some homogeneous spaces. We use harmonic analysis on the homogeneous spaces and representation theory of compact as well as noncompact reductive groups.

• The paper extends Whitney's classical extension problem to symmetric spaces by adapting harmonic analysis and Lie group representation theory.
• It establishes both approximative and exact extension results using explicit formulas, including Flensted-Jensen functions and holomorphic discrete series.
• The findings have practical implications for computational interpolation methods and contribute to advances in symmetry-aware AI models.

Whitney Extension Theorems on Symmetric Spaces: An Example

Introduction

The paper authored by Birgit Speh and Peter Vang Uttenthal addresses Whitney extension theorems within the context of symmetric spaces, specifically targeting homogeneous spaces such as $X(p, q)_{\pm}$. The primary focus is on extending the results originally conceptualized by H. Whitney in 1934 for Euclidean spaces to symmetric spaces equipped with inherent symmetries. The authors leverage harmonic analysis and representation theory of reductive Lie groups to derive their results, offering not just theoretical insights but also explicit, computable formulas.

Problem Statement

H. Whitney's classical problem involves extending a function $f$ defined on a finite set of points in $\mathbb{R}^n$ to an analytic function on the entire space. C. Fefferman further advanced this work by demonstrating a sharp form of the Whitney extension theorem for finite sets in Euclidean space. This paper endeavors to generalize Whitney's extension theorem to symmetric spaces $X = G/H$ for reductive Lie groups $G$ and $H$. The primary goal is to determine an analytic function that either approximates or exactly matches function values at specified points in these symmetric spaces.

Main Results

Two sets of results are central to this paper: approximative Whitney extensions and exact Whitney extensions. Initially, the authors provide conditions under which a complex-valued function can approximate given data points within any desired epsilon margin. Subsequently, they derive exact solutions through algorithmic and analytic techniques, involving harmonic analysis and representation theory.

Approximation Theorems

For a symmetric space $X(p,2)_{-}$, the authors show the existence of an analytic, square-integrable function $f_n$ such that:

$$|f_n(x^{(i)}) - y^{(i)}| < \varepsilon \quad \text{for all} \quad 1 \leq i \leq n.$$

Here, $f_n$ is generated as a linear combination of Flensted-Jensen functions, a form of eigenfunctions related to invariant differential operators on symmetric spaces.

Exact Extension Theorems

The paper also extends to exact Whitney extensions:

$$f_n(x^{(i)}) = y^{(i)} \quad \text{for all} \quad 1 \leq i \leq n.$$

This is achieved by leveraging the holomorphic structure and holomorphic discrete series representations inherent in the symmetric spaces considered. The exact solution involves constructing functional values that match the specified data points precisely, facilitated through the representation theory of noncompact Lie groups.

Implications and Future Directions

Practical Implications

The derivation of explicit formulas for Whitney functions has significant implications for computational applications, particularly in data science and machine learning where symmetry and group-invariance are pivotal. Specifically, this work can be influential for tasks involving interpolation and smoothing in high-dimensional spaces with inherent symmetries.

Theoretical Implications

From a theoretical standpoint, this paper extends the field of Whitney's extensions to new classes of spaces, providing a robust framework that could potentially be generalized to other forms of semisimple symmetric spaces. The methods outlined could serve as a precursor to understanding extensions in even more complex spaces equipped with rich symmetrical properties.

Speculations for AI Developments

In the field of AI, particularly in the development of equivariant neural networks and symmetry-aware algorithms, the methodology for extending functions on symmetric spaces could inspire architectural innovations. By integrating the representation of data points through Lie groups and symmetric spaces, the robustness and accuracy of AI models can be improved for applications necessitating high symmetry, such as molecular simulations and theoretical physics.

Conclusion

This paper offers a comprehensive extension of Whitney's extension theorems into symmetric spaces, presenting both approximate and exact forms of extension using sophisticated tools from harmonic analysis and representation theory. By ensuring the availability of explicit formulas and introducing effective algorithms, the research bridges the theoretical advancements from Whitney's classical problem into practical computational paradigms on symmetric spaces. The continuation of this analysis may open new avenues in both theoretical mathematics and applied AI research.