Displaying similar documents to “Sequence space representations for zero-solutions of convolution equations on ultradifferentiable functions of Roumieu type”

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 ∈ ε ω ( I ) ' with s u p p ( μ ) = 0 one can define the convolution operator T μ : ε ω ( I ) → ε ω ( I ) , T μ ( f ) ( x ) : = ⟨ μ , f ( x - · ) ⟩ . 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 μ ^ of μ.

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 ∈ ℇ ( ω ) ' ( ℝ n ) the surjectivity of the convolution operator T μ : ℇ ( ω ) ( Ω 1 ) → 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 ) → 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).