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

S. Melikhov; Siegfried Momm

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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...

Continuous linear right inverses for convolution operators in spaces of real analytic functions

Michael Langenbruch (1994)

Studia Mathematica

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We determine the convolution operators $T_{μ} : = μ *$ on the real analytic functions in one variable which admit a continuous linear right inverse. The characterization is given by means of a slowly decreasing condition of Ehrenpreis type and a restriction of hyperbolic type on the location of zeros of the Fourier transform μ̂(z).

The cancellation law for inf-convolution of convex functions

Dariusz Zagrodny (1994)

Studia Mathematica

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Conditions under which the inf-convolution of f and g $f □ g (x) : = i n f_{y + z = x} (f (y) + g (z))$ has the cancellation property (i.e. f □ h ≡ g □ h implies f ≡ g) are treated in a convex analysis framework. In particular, we show that the set of strictly convex lower semicontinuous functions $f : X \to ℝ \cup + \infty$ on a reflexive Banach space such that $l i m_{∥ x ∥ \to \infty} f (x) / ∥ x ∥ = \infty$ constitutes a semigroup, with inf-convolution as multiplication, which can be embedded in the group of its quotients.

Surjective convolution operators on spaces of distributions.

Leonhard Frerick, Jochen Wengenroth (2003)

RACSAM

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We review recent developments in the theory of inductive limits and use them to give a new and rather easy proof for Hörmander?s characterization of surjective convolution operators on spaces of Schwartz distributions.

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

B. A. Taylor, R. Meise, Dietmar Vogt (1990)

Annales de l'institut Fourier

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Solving a problem of L. Schwartz, those constant coefficient partial differential operators $P (D)$ are characterized that admit a continuous linear right inverse on $ℰ (Ω)$ or $𝒟^{'} (Ω)$ , $Ω$ an open set in $R^{n}$ . For bounded $Ω$ with $C^{1}$ -boundary these properties are equivalent to $P (D)$ being very hyperbolic. For $Ω = R^{n}$ they are equivalent to a Phragmen-Lindelöf condition holding on the zero variety of the polynomial $P$ .

Surjectivity of convolution operators on spaces of ultradifferentiable functions of Roumieu type

Thomas Meyer (1997)

Studia Mathematica

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Let $ε_{ω} (I)$ denote the space of all ω-ultradifferentiable functions of Roumieu type on an open interval I in ℝ. In the special case ω(t) = t we get the real-analytic functions on I. For $μ \in ε_{ω} {(I)}^{'}$ with $s u p p (μ) = 0$ one can define the convolution operator $T_{μ} : ε_{ω} (I) \to ε_{ω} (I)$ , $T_{μ} (f) (x) : = ⟨ μ, f (x - \cdot) ⟩$ . We give a characterization of the surjectivity of $T_{μ}$ for quasianalytic classes $ε_{ω} (I)$ , where I = ℝ or I is an open, bounded interval in ℝ. This characterization is given in terms of the distribution of zeros of the Fourier Laplace transform $\hat{μ}$ of μ.

Continuation of holomorphic solutions to convolution equations in complex domains

Ryuichi Ishimura, Jun-ichi Okada, Yasunori Okada (2000)

Annales Polonici Mathematici

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For an analytic functional $S$ on $ℂ^{n}$ , we study the homogeneous convolution equation S * f = 0 with the holomorphic function f defined on an open set in $ℂ^{n}$ . We determine the directions in which every solution can be continued analytically, by using the characteristic set.

Some remarks on convolution equations

C. A. Berenstein, M. A. Dostal (1973)

Annales de l'institut Fourier

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Using a description of the topology of the spaces $E^{'} (Ω)$ ( $Ω$ open convex subset of $R^{n}$ ) via the Fourier transform, namely their analytically uniform structures, we arrive at a formula describing the convex hull of the singular support of a distribution $T$ , $T \in E^{'}$ . We give applications to a class of distributions $T$ satisfying $cv. sing. supp. S * T = cv. sing. supp. S + cv. sing. supp. T$ for all $S \in E^{'}$ .

Convolution operators in infinite dimension.

Nguyen Van Khue, Nguyen Dinh Sang (1994)

Portugaliae Mathematica

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Displaying similar documents to “Solution operators for convolution equations on the germs of analytic functions on compact convex sets in ℂ N ”

Displaying similar documents to “Solution operators for convolution equations on the germs of analytic functions on compact convex sets in $ℂ^{N}$ ”