Kolmogorov diameters in solution spaces of systems of partial differential equations
type: Item
instance of: scholarly article
title: Kolmogorov diameters in solution spaces of systems of partial differential equations
zbMATH Open document ID: 0616.35010
DOI: 10.1007/BF01174010
author: Michael Langenbruch
published in: Manuscripta Mathematica
publication date: 1985-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/155088
review text: The functional dimension of the space of smooth solutions on \({\mathbb{R}}^ n\) of partial differential equations is a weak linear topological invariant, which in general does not have entirely adequate separation properties. In this paper the author considers the stronger linear topological invariant of diametral dimension of solution spaces of partial differential equations. Specifically if \(E_ p(0),0\subset {\mathbb{R}}^ n\) denotes the solutions of P(D) a system of partial differential equations with constant coefficients in a localisable analytically uniform space E then the relative Kolmogorov diameters of the neighbourhoods of 0 in \(E_ p(0)\) are estimated from above and below using a principle due to Ehrenpreis. The diametrial dimension of \(E_ p(0)\) is calculated and it is shown that in special cases \(E_ p({\mathbb{R}}^ n)\) and \(E_ p(0)\) are not isomorphic for 0 bounded.
Mathematics Subject Classification ID: 35E20
Mathematics Subject Classification ID: 35B40
Mathematics Subject Classification ID: 35B65
zbMATH DE Number: 3998311
zbMATH Keywords: functional dimension
zbMATH Keywords: smooth solutions
zbMATH Keywords: topological invariant
zbMATH Keywords: diametral dimension
zbMATH Keywords: localisable analytically uniform space
zbMATH Keywords: Kolmogorov diameters
reviewed by: G. F. Roach
MaRDI profile type: Publication
cites work: Linear partial differential operators and generalized distributions
cites work: Q5724974
cites work: Localizable Analytically Uniform Spaces and the Fundamental Principle
cites work: Q5512342
cites work: Q3843833
cites work: Q3918541
cites work: ON THE FUNCTIONAL DIMENSION OF THE SOLUTION SPACE OF HYPOELLIPTIC EQUATIONS
cites work: Q5673937
cites work: Q4165767
cites work: Q5577934
cites work: Differentiability and growth of solutions of partial differential equations
cites work: On the functional dimension of solution spaces of hypoelliptic partial differential operators
cites work: Bases in solution sheaves of systems of partial differential equations.
cites work: Q3889437
cites work: Eine Charakterisierung der Potenzreihenraeume von endlichem Typ und ihre Folgerungen
cites work: Charakterisierung der Unterraüme und Quotienteräume der nuklearen stabilen Potenzreihenraüme von unendlichem Typ
cites work: On the functional dimension of the space of solutions of partial differential equations
description: scientific article; zbMATH DE number 3998311