Approximations of Symmetric Functions on Banach Spaces with Symmetric Bases

Mariia Martsinkiv,Andriy Zagorodnyuk

doi:10.3390/sym13122318

Abstract

This paper is devoted to studying approximations of symmetric continuous functions by symmetric analytic functions on a Banach space X with a symmetric basis. We obtain some positive results for the case when X admits a separating polynomial using a symmetrization operator. However, even in this case, there is a counter-example because the symmetrization operator is well defined only on a narrow, proper subspace of the space of analytic functions on X. For X=c0, we introduce ε-slice G-analytic functions that have a behavior similar to G-analytic functions at points x∈c0 such that all coordinates of x are greater than ε, and we prove a theorem on approximations of uniformly continuous functions on c0 by ε-slice G-analytic functions.

Highlights

Introduction and PreliminariesSymmetric functions of infinitely many variables naturally appear in problems of statistical mechanics, particle physics, deep learning models, neural networks, and other brunches of knowledge that proceed with big amounts of data that do not depend on ordering
The paper is devoted to studying approximations of symmetric continuous functions by symmetric analytic functions on Banach spaces with symmetric bases
For many cases, such an approximation is impossible, even if every continuous function on the Banach space X can be approximated by analytic ones

Summary

Introduction and Preliminaries

Symmetric functions of infinitely many variables naturally appear in problems of statistical mechanics, particle physics, deep learning models, neural networks, and other brunches of knowledge that proceed with big amounts of data that do not depend on ordering (see, e.g., [1,2]). Due to Kurzweil’s theorem [5], if X admits a separating polynomial, every continuous function on X can be approximated by analytic functions uniformly on the whole space X. This result was extended to the case when X admits a separating analytic function by Boiso and Hajek in [6]. We discuss the possibility of approximating symmetric continuous functions on a Banach space with symmetric basis by some special symmetric functions (in particular, polynomials or analytic functions). Since c0 does not support symmetric polynomials, we introduce symmetric ε-slice polynomials and ε-slice G-analytic functions and prove a theorem about approximation by such functions

Kurzweil’s Approximation and the Symmetrization Operator

Lipschitz Symmetric Functions

Approximations of Symmetric Functions on c0

Conclusions