Convolution equations in domains of \(\mathbb{C}^ n\)
type: Item
instance of: scholarly article
title: Convolution equations in domains of \(\mathbb{C}^ n\)
zbMATH Open document ID: 0794.32004
DOI: 10.1007/BF02384343
author: Ragnar Sigurdsson
published in: Arkiv för Matematik
publication date: 1992-09-27T00:00:00Z
review text: The author deals with the study of convolution equations of the form \(\mu*u=f\), where \(\mu \in A'(\mathbb{C}^ n)\) is an analytic functional and \(u \in A(X_ 1)\), \(f \in A (X_ 2)\) are analytic functions in open subsets \(X_ 1\) and \(X_ 2\) of \(\mathbb{C}^ n\). Introducing the concept of \(\mu\)-convexity for carriers of a pair \((X_ 1, X_ 2)\) of open subsets of \(C^ n\), the growth properties of plurisubharmonic and analytic functions in \(\mathbb{C}^ n\), as well as the growth properties of Fourier- Laplace transforms are emphasized. Thus, the author is concerned with the problem of necessary and sufficient conditions for existence of solutions \(u \in A(X_ 1)\) of \(\mu*u=f\) for any \(f \in A (X_ 2)\) and with an approximation problem: to decide if the space \(E_ \mu\), consisting of all linear combinations of exponential solutions of \(\mu*u=0\) in \(\mathbb{C}^ n\), is dense in the space \(N(X_ 1, X_ 2)\) of all solutions \(u \in A(X_ 1)\) of \(\mu*u=0\) in \(X_ 2\).
Mathematics Subject Classification ID: 32A25
Mathematics Subject Classification ID: 44A35
Mathematics Subject Classification ID: 46F15
Mathematics Subject Classification ID: 32A45
zbMATH DE Number: 60554
zbMATH Keywords: convexity
zbMATH Keywords: convolution equations
zbMATH Keywords: analytic functional
zbMATH Keywords: analytic functions
zbMATH Keywords: plurisubharmonic
zbMATH Keywords: Fourier-Laplace transforms
reviewed by: Petre P. Teodorescu
MaRDI profile type: Publication
cites work: ON THE ASYMPTOTIC BEHAVIOR OF SUBHARMONIC FUNCTIONS OF FINITE ORDER
cites work: Q3665402
cites work: Solution of Some Problems of Division. Part IV. Invertible and Elliptic Operators
cites work: Q5590963
cites work: On the range of convolution operators
cites work: Supports and singular supports of convolutions
cites work: Convolution equations in convex domains
cites work: Q5512342
cites work: Q3671563
cites work: Limit sets of plurisubharmonic functions.
cites work: INVARIANT SUBSPACES OF ANALYTIC FUNCTIONS. I. SPECTRAL ANALYSIS ON CONVEX REGIONS
cites work: Q3708270
cites work: Existence et approximation des solutions des équations aux dérivées partielles et des équations de convolution
cites work: Q3275209
cites work: Sur les fonctionnelles analytiques et la transformation de Fourier-Borel
cites work: Équations différentielles d'ordre infini
cites work: Q3764643
cites work: Convolution equations in spaces of functions holomorphic in convex domains and on convex compacta in \(C^n\)
cites work: ON EPIMORPHICITY OF A CONVOLUTION OPERATOR IN CONVEX DOMAINS IN $ \mathbf{C}^l$
cites work: AN EQUATION OF CONVOLUTION TYPE IN TUBE DOMAINS OFC2
cites work: ON A UNIQUENESS THEOREM IN THE THEORY OF FUNCTIONS OF SEVERAL COMPLEX VARIABLES, AND HOMOGENEOUS EQUATIONS OF CONVOLUTION TYPE IN TUBE DOMAINS OFCn
cites work: Convolution equations in multidimensional spaces
cites work: Q5613420
cites work: Growth properties of analytic and plurisubharmonic functions of finite order.
cites work: Q3688646
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description: scientific article; zbMATH DE number 60554