Density of polynomial elements in invariant subspaces of entire functions of exponential type
type: Item
instance of: scholarly article
title: Density of polynomial elements in invariant subspaces of entire functions of exponential type
zbMATH Open document ID: 1058.32002
DOI: 10.1023/B:MATN.0000023317.50105.73
author: V. M. Trutnev
published in: Mathematical Notes
publication date: 2004-12-06T00:00:00Z
review text: When studying the approximation problem for arbitrary solutions (in different classes of functions) of a differential equation with constant coefficients by polynomial solutions, Malgrange found a necessary and sufficient condition for this approximation, namely, the irreducible components of the characteristic polynomial of the differential equation must vanish at the origin. Later on, Palamodov proved for systems of differential equations that every solution can be approximated by exponential polynomials of the form \(p(z)\exp(z,\zeta)\), \(\zeta\in B\), where the set \(B\supset\mathbb C^n\) has nonempty intersection with every manifold in the corresponding family of Noetherian operators and manifolds. The objective of the present paper is the extension of the Malgrange and Palamodov results to translation invariant subspaces in the space of entire functions of exponential type of several complex variables. Conditions are presented under which the set of exponential polynomials is dense in such a space. In particular, necessary and sufficient conditions for the polynomial elements to be dense in invariant subspaces are obtained.
Mathematics Subject Classification ID: 32A15
Mathematics Subject Classification ID: 32A37
Mathematics Subject Classification ID: 46E15
Mathematics Subject Classification ID: 46J15
Mathematics Subject Classification ID: 47A15
zbMATH DE Number: 2121367
zbMATH Keywords: entire function of exponential type
zbMATH Keywords: translation invariant subspaces
zbMATH Keywords: exponential polynomials
zbMATH Keywords: Laplace transform of functional
zbMATH Keywords: Runge domain
zbMATH Keywords: characteristic analytic set
zbMATH Keywords: spectral synthesis
zbMATH Keywords: holomorphic function
zbMATH Keywords: Noetherian ring
zbMATH Keywords: primary ideal
zbMATH Keywords: annihilator ideal
reviewed by: Eleonora A. Storozhenko
MaRDI profile type: Publication
full work available at URL: b:matn.0000023317.50105.73
OpenAlex ID: W2090518026
description: scientific article; zbMATH DE number 2121367