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Andreas Debrouwere

Publications and source records attributed to Andreas Debrouwere.

48 records · Page 3Linked to original sources
arXiv2019

Quasinormable $C_0$-groups and translation-invariant Fréchet spaces of type $\mathcal{D}_E$

Let $E$ be a locally convex Hausdorff space satisfying the convex compact property and let $(T_x)_{x \in \mathbb{R}^d}$ be a locally equicontinuous $C_0$-group of linear continuous operators on $E$. In this article, we show that if $E$ is quasinormable, then the space of smooth vectors in $E$ associated to $(T_x)_{x \in \mathbb{R}^d}$ is also quasinormable. In particular, we obtain that the space of smooth vectors associated to a $C_0$-group on a Banach space is always quasinormable. As an application, we show that the translation-invariant Fréchet spaces of smooth functions of type $\mathcal{D}_E$ [8] are quasinormable, thereby settling the question posed in [8, Remark 7]. Furthermore, we show that $\mathcal{D}_E$ is not Montel if $E$ is a solid translation-invariant Banach space of distributions [10]. This answers the question posed in [8, Remark 6] for the class of solid translation-invariant Banach spaces of distributions.

math.FA↗
arXiv2019

A linear topological invariant for spaces of quasianalytic functions of Roumieu type

We show that the spaces $\mathcal{E}_{\{ω\}}(Ω)$ of ultradifferentiable functions of Roumieu type satisfy the dual interpolation estimate for small theta, where $ω$ is a quasianalytic weight function and $Ω$ is an arbitrary open subset of $\mathbb{R}^d$. This result was previously shown by Bonet and Domański [2] under the additional assumptions that $Ω$ is convex and $ω$ satisfies the condition $(α_1)$. In particular, our work solves Problem 9.7 in [1].

math.FA↗
arXiv2019

Injectivity and surjectivity of the Stieltjes moment mapping in Gelfand-Shilov spaces

The Stieltjes moment problem is studied in the framework of general Gelfand-Shilov spaces defined via weight sequences. We characterize the injectivity and surjectivity of the Stieltjes moment mapping, sending a function to its sequence of moments, in terms of growth conditions for the defining weight sequence. Finally, a related moment problem at the origin is studied.

math.FA↗
arXiv2018

On weighted inductive limits of spaces of ultradifferentiable functions and their duals

In the first part of this paper we discuss the completeness of two general classes of weighted inductive limits of spaces of ultradifferentiable functions. In the second part we study their duals and characterize these spaces in terms of the growth of convolution averages of their elements. This characterization gives a canonical way to define a locally convex topology on these spaces and we give necessary and sufficient conditions for them to be ultrabornological. In particular, our results apply to spaces of convolutors for Gelfand-Shilov spaces.

math.FA↗
arXiv2017

Generalized function algebras containing spaces of periodic ultradistributions

We construct differential algebras in which spaces of (one-dimensional) periodic ultradistributions are embedded. By proving a Schwartz impossibility type result, we show that our embeddings are optimal in the sense of being consistent with the pointwise multiplication of ordinary functions. In particular, we embed the space of hyperfunctions on the unit circle into a differential algebra in such a way that the multiplication of real analytic functions on the unit circle coincides with their pointwise multiplication. Furthermore, we introduce a notion of regularity in our newly defined algebras and show that an embedded ultradistribution is regular if and only if it is an ultradifferentiable function.

math.FA↗
arXiv2017

Sheaves of nonlinear generalized function spaces

We provide a framework for the construction of diffeomorphism invariant sheaves of nonlinear generalized functions spaces. As an application, global algebras of generalized functions for distributions on manifolds and diffeomorphism invariant algebras of generalized functions for ultradistributions are constructed.

math.FA↗
arXiv2017

On the non-triviality of certain spaces of analytic functions. Hyperfunctions and ultrahyperfunctions of fast growth

We study function spaces consisting of analytic functions with fast decay on horizontal strips of the complex plane with respect to a given weight function. Their duals, so called spaces of (ultra)hyperfunctions of fast growth, generalize the spaces of Fourier hyperfunctions and Fourier ultrahyperfunctions. An analytic representation theory for their duals is developed and applied to characterize the non-triviality of these function spaces in terms of the growth order of the weight function. In particular, we show that the Gelfand-Shilov spaces of Beurling type $\mathcal{S}^{(p!)}_{(M_p)}$ and Roumieu type $\mathcal{S}^{\{p!\}}_{\{M_p\}}$ are non-trivial if and only if $$ \sup_{p \geq 2}\frac{(\log p)^p}{h^pM_p} < \infty, $$ for all $h > 0$ and some $h > 0$, respectively. We also study boundary values of holomorphic functions in spaces of ultradistributions of exponential type, which may be of quasianalytic type.

math.FA↗
arXiv2016

Optimal embeddings of ultradistributions into differential algebras

We construct embeddings of spaces of non-quasianalytic ultradistributions into differential algebras enjoying optimal properties in view of a Schwartz type impossibility result, also shown in this article. We develop microlocal analysis in theses algebras consistent with the microlocal analysis in the corresponding spaces of ultradistributions.

math.FA↗
arXiv2016

Discrete characterizations of wave front sets of Fourier-Lebesgue and quasianalytic type

We obtain discrete characterizations of wave front sets of Fourier-Lebesgue and quasianalytic type. It is shown that the microlocal properties of an ultradistribution can be obtained by sampling the Fourier transforms of its localizations over a lattice in $\mathbb{R}^{d}$. In particular, we prove the following discrete characterization of the analytic wave front set of a distribution $f\in\mathcal{D}'(Ω)$. Let $Λ$ be a lattice in $\mathbb{R}^{d}$ and let $U$ be an open convex neighborhood of the origin such that $U\capΛ^{*}=\{0\}$. The analytic wave front set $WF_{A}(f)$ coincides with the complement in $Ω\times(\mathbb{R}^{d}\setminus\{0\})$ of the set of points $(x_0,ξ_0)$ for which there are an open neighborhood $V\subset Ω\cap (x_0+U)$ of $x_0$, an open conic neighborhood $Γ$ of $ξ_0$, and a bounded sequence $(f_p)_{p \in \mathbb{N}}$ in $\mathcal{E}'(Ω\cap (x_0+U))$ with $f_p= f$ on $V$ such that for some $h > 0$ \[ \sup_{μ\in Γ\cap Λ} |\widehat{f_p} (μ)| |μ|^p \leq h^{p+1}p!\:, \qquad \forall p \in \mathbb{N}. \]

math.AP↗