Characterization of the linear partial differential operators with constant coefficients that admit a continuous linear right inverse

B. A. Taylor; R. Meise; Dietmar Vogt

Displaying similar documents to “Characterization of the linear partial differential operators with constant coefficients that admit a continuous linear right inverse”

Extension and lacunas of solutions of linear partial differential equations

Uwe Franken, Reinhold Meise (1996)

Annales de l'institut Fourier

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Let $K \subset Q$ be compact, convex sets in $ℝ^{n}$ with $\overset{\circ}{K} \neq \emptyset$ and let $P (D)$ be a linear, constant coefficient PDO. It is characterized in various ways when each zero solution of $P (D)$ in the space $ℰ (K)$ of all $C^{\infty}$ -functions on $K$ extends to a zero solution in $ℰ (Q)$ resp. in $ℰ (ℝ^{n})$ . The most relevant characterizations are in terms of Phragmén-Lindelöf conditions on the zero variety of $P$ in $ℂ^{n}$ and in terms of for $P (D)$ with lacunas.

Solution operators for convolution equations on the germs of analytic functions on compact convex sets in $ℂ^{N}$

S. Melikhov, Siegfried Momm (1995)

Studia Mathematica

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$G \subset ℂ^{N}$ is compact and convex it is known for a long time that the nonzero constant coefficients linear partial differential operators (of finite or infinite order) are surjective on the space of all analytic functions on G. We consider the question whether solutions of the inhomogeneous equation can be given in terms of a continuous linear operator. For instance we characterize those sets G for which this is always the case.

Localizations of partial differential operators and surjectivity on real analytic functions

Michael Langenbruch (2000)

Studia Mathematica

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Let P(D) be a partial differential operator with constant coefficients which is surjective on the space A(Ω) of real analytic functions on an open set $Ω \subset ℝ^{n}$ . Then P(D) admits shifted (generalized) elementary solutions which are real analytic on an arbitrary relatively compact open set ω ⊂ ⊂ Ω. This implies that any localization $P_{m, Θ}$ of the principal part $P_{m}$ is hyperbolic w.r.t. any normal vector N of ∂Ω which is noncharacteristic for $P_{m, Θ}$ . Under additional assumptions $P_{m}$ must be locally hyperbolic. ...

On the range of convolution operators on non-quasianalytic ultradifferentiable functions

Jóse Bonet, Antonio Galbis, R. Meise (1997)

Studia Mathematica

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Let $ℇ_{(ω)} (Ω)$ denote the non-quasianalytic class of Beurling type on an open set Ω in $ℝ^{n}$ . For $μ \in ℇ_{(ω)}^{'} (ℝ^{n})$ the surjectivity of the convolution operator $T_{μ} : ℇ_{(ω)} (Ω_{1}) \to ℇ_{(ω)} (Ω_{2})$ is characterized by various conditions, e.g. in terms of a convexity property of the pair $(Ω_{1}, Ω_{2})$ and the existence of a fundamental solution for μ or equivalently by a slowly decreasing condition for the Fourier-Laplace transform of μ. Similar conditions characterize the surjectivity of a convolution operator $S_{μ} : D_{ω}^{'} (Ω_{1}) \to D_{ω}^{'} (Ω_{2})$ between ultradistributions of Roumieu type whenever...

On the closure of spaces of sums of ridge functions and the range of the $X$ -ray transform

Jan Boman (1984)

Annales de l'institut Fourier

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For $a \in R^{n} ∖ {0}$ and $Ω$ an open bounded subset of $R^{n}$ definie $L^{p} (Ω, a)$ as the closed subset of $L^{p} (Ω)$ consisting of all functions that are constant almost everywhere on almost all lines parallel to $a$ . For a given set of directions $a^{ν} \in R^{n} ∖ {0}$ , $ν = 1, ..., m$ , we study for which $Ω$ it is true that the vector space $(*) L^{p} (Ω, a^{1}) + \dots + L^{p} (Ω, a^{m}) is a closed subspace of L^{p} (Ω) .$ This problem arizes naturally in the study of image reconstruction from projections (tomography). An essentially equivalent problem is to decide whether a certain matrix-valued differential operator...

On Bárány's theorems of Carathéodory and Helly type

Ehrhard Behrends (2000)

Studia Mathematica

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The paper begins with a self-contained and short development of Bárány’s theorems of Carathéodory and Helly type in finite-dimensional spaces together with some new variants. In the second half the possible generalizations of these results to arbitrary Banach spaces are investigated. The Carathéodory-Bárány theorem has a counterpart in arbitrary dimensions under suitable uniform compactness or uniform boundedness conditions. The proper generalization of the Helly-Bárány theorem reads...

On absolutely representing systems in spaces of infinitely differentiable functions

Yu. Korobeĭnik (2000)

Studia Mathematica

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The main part of the paper is devoted to the problem of the existence of absolutely representing systems of exponentials with imaginary exponents in the spaces $C^{\infty} (G)$ and $C^{\infty} (K)$ of infinitely differentiable functions where G is an arbitrary domain in $ℝ^{p}$ , p≥1, while K is a compact set in $ℝ^{p}$ with non-void interior K̇ such that $\bar{K} ̇ = K$ . Moreover, absolutely representing systems of exponents in the space H(G) of functions analytic in an arbitrary domain $G \subseteq ℂ^{p}$ are also investigated.

A sufficient condition for existence of real analytic solutions of P.D.E. with constant coefficients, in open sets of $ℝ^{2}$

Giuseppe Zampieri (1980)

Rendiconti del Seminario Matematico della Università di Padova

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Theorems of Krein Milman type for certain convex sets of functions operators

Robert R. Phelps (1970)

Annales de l'institut Fourier

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Sufficient conditions are given in order that, for a bounded closed convex subset $B$ of a locally convex space $E$ , the set $C (X, B)$ of continuous functions from the compact space $X$ into $B$ , is the uniformly closed convex hull in $C (X, E)$ of its extreme points. Applications are made to the unit ball of bounded (or compact, or weakly compact) operators from certain Banach spaces into $C (X)$ .