Topology of the space of locally separable functions
type: Item
instance of: scholarly article
title: Topology of the space of locally separable functions
zbMATH Open document ID: 1478.26011
author: Hassan Boualem
author: Robert Brouzet
published in: Topology and its Applications
publication date: 2021-07-28T00:00:00Z
review text: The set \(\mathcal S\) of real \(C^\infty\)-functions of \(n\) variables which are of the form \(H(x)=u_1(x)+\dots+u_n(x)\) on some neighbourhood \(V\subset \mathbb R^n\) of the origin for some diffeomorphism \(u:V \to W\) such that \(u(0)=0\), is investigated. It is shown that \(\mathcal S\) contains an open dense subset of the Frechet space \(C^\infty(U,\mathbb R)\) for any fixed open neighbourhood \(U\) of the origin. The interior of \(\mathcal S\) is studied in more detail. Using the classification of the functions by the properties of the \(k\)-homogeneous polynomials \(T_k (H)\) of the Taylor expansion of \(H\) at the origin, two main results on \(\mathcal S\) are deduced. For \(n=2\), the set \(\text{Int} (\mathcal S)\) is fully described as the union of open sets of functions -- \(O_1\) of those with nonzero \(T_1(H)\), \(O_2\) of those with nonzero \(T_2(H)\), and a particular subset of analalogically defined \(O_3\). For \(n\ge 3\), just two inclusions approximating \(\text{Int} (\mathcal S)\) are proved. Namely, \(O_1\cup O_2(n)\cup O_2(n-1)\subset \text{Int} (\mathcal S) \subset O_1\cup O_2(n)\cup \dots \cup O_2(r_n)\) for uniquely defined integer \(r_n<n\). Here, \(O_2(r)\) is the subset of elements of \(O_2\) for which the rank of the Hessian at the origin is \(r\).
Mathematics Subject Classification ID: 26B40
Mathematics Subject Classification ID: 26B10
Mathematics Subject Classification ID: 26E10
Mathematics Subject Classification ID: 57M60
zbMATH DE Number: 7375602
zbMATH Keywords: \(C^\infty\)-functions of \(n\) variables
zbMATH Keywords: locally separated functions
zbMATH Keywords: Taylor expansion
zbMATH Keywords: \(k\)-jet
reviewed by: Petr Holický
MaRDI profile type: Publication
full work available at URL: j.topol.2021.107728
OpenAlex ID: W3165251700
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DOI: 10.1016/J.TOPOL.2021.107728
description: scientific article; zbMATH DE number 7375602