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

Thomas Meyer

Studia Mathematica (1997)

  • Volume: 125, Issue: 2, page 101-129
  • ISSN: 0039-3223

Abstract

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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.

How to cite

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Meyer, Thomas. "Surjectivity of convolution operators on spaces of ultradifferentiable functions of Roumieu type." Studia Mathematica 125.2 (1997): 101-129. <http://eudml.org/doc/216426>.

@article{Meyer1997,
abstract = {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 $μ ∈ ε_\{\{ω\}\}(I)^\{\prime \}$ with $supp(μ) = \{0\}$ 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 $\widehat\{μ\}$ of μ.},
author = {Meyer, Thomas},
journal = {Studia Mathematica},
keywords = {convolution operator; ultradifferentiable functions},
language = {eng},
number = {2},
pages = {101-129},
title = {Surjectivity of convolution operators on spaces of ultradifferentiable functions of Roumieu type},
url = {http://eudml.org/doc/216426},
volume = {125},
year = {1997},
}

TY - JOUR
AU - Meyer, Thomas
TI - Surjectivity of convolution operators on spaces of ultradifferentiable functions of Roumieu type
JO - Studia Mathematica
PY - 1997
VL - 125
IS - 2
SP - 101
EP - 129
AB - 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 $μ ∈ ε_{{ω}}(I)^{\prime }$ with $supp(μ) = {0}$ 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 $\widehat{μ}$ of μ.
LA - eng
KW - convolution operator; ultradifferentiable functions
UR - http://eudml.org/doc/216426
ER -

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