TDI-subspaces of \(C(\mathbb{R}^ d)\) and some density problems from neural networks
type: Item
instance of: scholarly article
title: TDI-subspaces of \(C(\mathbb{R}^ d)\) and some density problems from neural networks
zbMATH Open document ID: 0858.41023
author: Allan Pinkus
published in: Journal of Approximation Theory
publication date: 1996-08-05T00:00:00Z
review text: Space \(W\subset C(\mathbb{R}^d)\) is said to be TDI if it is invariant under actions of the group generated by translations and dilations (in each coordinates) of \(\mathbb{R}^d\). Let \(M_f\) be the closure of the smallest TDI-space which contains \(f\). Theorem 1. \(M_f\neq C(\mathbb{R}^d)\Leftrightarrow\) there is \(\alpha\in\mathbb{Z}^d_+\) such that \(D^\alpha f=0\) (in the weak sense). Let \(A_\alpha:=\overline{\text{span}}\{x^\beta; \beta\leq\alpha\}\); here some coordinates of \(\alpha\) may be infinite but all \(\beta\in\mathbb{Z}^d_+\). Theorem 2. Every TDI-space is a finite sum of the \(A_\alpha\).
Mathematics Subject Classification ID: 41A63
Mathematics Subject Classification ID: 41A30
Mathematics Subject Classification ID: 92B20
zbMATH DE Number: 894208
zbMATH Keywords: dilation invariant space
zbMATH Keywords: density problem
reviewed by: Yuri A. Brudnyi
MaRDI profile type: Publication
full work available at URL: jath.1996.0042
OpenAlex ID: W2052742287
DOI: 10.1006/JATH.1996.0042
description: scientific article; zbMATH DE number 894208