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Lenny Neyt

Publications and source records attributed to Lenny Neyt.

At least 19 recordsLinked to original sources
arXiv2026

Spectral characterizations of stable operator semigroups

We introduce the notion of local pseudofunction spectrum $σ_{PF}(A)$ for the infinitesimal generator $A$ of a bounded $C_0$-semigroup $\mathcal{T} = (T(t))_{t \geq 0}$ on a Banach space $X$ and show it is the right spectral concept to deliver a full characterization of the strong stability of $\mathcal{T}$: $$ \forall x \in X : ~ \lim_{t \to \infty} \| T(t) x \|_X = 0 \quad \Longleftrightarrow \quad σ_{PF}(A) = \varnothing. $$ We demonstrate how this yields a quick proof of the well-known Arendt-Batty-Lyubich-Vũ theorem and establish novel stability results through local range density conditions for semigroups whose local pseudofunction spectra are a null subset of the imaginary axis. We also obtain similar stability characterization theorems for individual orbits and for semi-uniform stability. As an application of our results, we provide spectral characterizations of almost periodic $C_0$-semigroups with countable spectrum. In addition, we prove optimal Tauberian theorems of Katznelson-Tzafriri type and discuss connections with Wiener kernels.

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arXiv2025

Optimal decay of semi-uniformly stable operator semigroups with empty spectrum

We show that it is impossible to quantify the decay rate of a semi-uniformly stable operator semigroup based on sole knowledge of the spectrum of its infinitesimal generator. More precisely, given an arbitrary positive function $r$ vanishing at $\infty$, we construct a Banach space $X$ and a bounded semigroup $ (T(t))_{t \geq 0}$ of operators on it whose infinitesimal generator $A$ has empty spectrum $σ(A)=\varnothing$, but for which, for some $x \in X$, $$ \limsup_{t\to\infty} \frac{\|T(t)A^{-1}x\|_{X}}{r(t)}=\infty. $$

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arXiv2025

Quantitative uncertainty principles for time-frequency Gaussian decay

For real symmetric positive definite matrices $A$ and $B$, we characterize when a function $f \in L^2(\mathbb{R}^d)$ satisfies \[ |f(x)| \lesssim e^{-(\frac12 - λ) \langle Ax, x\rangle} \quad \text{and} \quad |\widehat{f}(ξ)| \lesssim e^{-(\frac12 - λ) \langle Bξ, ξ\rangle} , \qquad \forall λ> 0 , \] or even more specified time-frequency decay estimates, in terms of the skewed Hermite series expansion of $f$. We also consider coordinate-wise time-frequency decay and determine when it becomes equivalent to the same bounds on the skewed Hermite coefficients.

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arXiv2025

Bornological LB-spaces and idempotent adjunctions

The notion of an LB-space was introduced by Grothendieck in his 1953 thèse, referring to a countable colimit of Banach spaces taken within the category of locally convex topological vector spaces, and refining prior work done by Dieudonné, Schwartz and Köthe. Recently, two different notions of `bornological LB-spaces' emerged: one, given by Stempfhuber, refers to countable colimits of Banach spaces as well, but now taken in the category of bornological vector spaces. The other one, given by Bambozzi, Ben-Bassat and Kremnizer, refers to bornologifications of regular LB-spaces, i.e., of LB-spaces in the Grothendieck sense having the additional property that every bounded subset of the colimit is contained and bounded in one of its Banach steps. In this note, we show that the two notions are distinct, but nevertheless closely related. This involves, in particular, an intimate study of the idempotent adjunction of the bornologification and topologification functors.

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arXiv2025

Weighted function spaces: convolutors, multipliers, and mollifiers

We study smooth function spaces of Gelfand-Shilov type, with global behavior governed through a translation-invariant Banach function space and localized via a weight function system. We clarify the roles of the translation-invariant Banach function space, convolution, and pointwise multiplication in connection with the weight function system. Our primary goal is to characterize these function spaces-as well as the corresponding convolutor and multiplier spaces-through mollification. For this purpose, we introduce the moment-wise decomposition factorization property for pairs of compactly supported smooth functions, and establish complete characterizations in terms of mollifications with these windows.

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arXiv2025

Sequence space representations of Beurling-Björck spaces via Gabor frames and Wilson bases

We establish sequence space representations of a broad class of Beurling-Björck spaces $\mathcal{S}^{(ω)}_{(η)}$ and $\mathcal{S}^{\{ω\}}_{\{η\}}$. We develop two different approaches: a non-constructive one based on Gabor frames and the structure theory of Fréchet spaces, and a constructive one using Wilson bases, under stronger assumptions on the defining weight functions $ω$ and $η$. As an application, we provide an isomorphic classification of the spaces $\mathcal{S}^{(ω)}_{(η)}$ and $\mathcal{S}^{\{ω\}}_{\{η\}}$ in terms of $ω$ and $η$. In particular, our results are applicable to the classical Gelfand-Shilov spaces $\mathcal{S}^μ_τ$ for $μ, τ\geq 1/2$ (non-constructive approach) and $μ, τ\geq 1$ (constructive approach).

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arXiv2025

On the inclusion relations between Gelfand-Shilov spaces

We study inclusion relations between Gelfand-Shilov type spaces defined via a weight (multi-)sequence system, a weight function system, and a translation-invariant Banach function space. We characterize when such spaces are included into one another in terms of growth relations for the defining weight sequence and function systems. Our general framework allows for a unified treatment of the Gelfand-Shilov spaces $\mathcal{S}^{[M]}_{[A]}$ (defined via weight sequences $M$ and $A$) and the Beurling-Björck spaces $\mathcal{S}^{[ω]}_{[η]}$ (defined via weight functions $ω$ and $η$).

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arXiv2025

On a space of functions with entire Laplace transforms and its connection with the optimality of the Ingham-Karamata theorem

We study approximation properties of the Fréchet space of all continuously differentiable functions $τ$ such that $τ'(x)=o(1)$ and such that their Laplace transforms admit entire extensions to $\mathbb{C}$. As an application, these approximation results are combined with the open mapping theorem to show the optimality theorem for the Ingham-Karamata Tauberian theorem.

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arXiv2024

Hermite expansions for spaces of functions with nearly optimal time-frequency decay

We establish Hermite expansion characterizations for several subspaces of the Fréchet space of functions on the real line satisfying \begin{equation*} |f(x)| \lesssim e^{-(\frac{1}{2} - λ) x^{2}} , \qquad | \widehat{f}(ξ)| \lesssim e^{-(\frac{1}{2} - λ) ξ^{2}} , \qquad \forall λ> 0 . \end{equation*} In particular, we extend and improve Fourier characterizations of the so-called proper Pilipović spaces obtained in [J. Funct. Anal. 284 (2023), 109724]. The main ingredients in our proofs are the Bargmann transform and some achieved optimal forms of the Phragmén-Lindelöf principle.

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arXiv2024

The vector-valued Stieltjes moment problem with general exponents

We characterize the sequences of complex numbers $(z_{n})_{n \in \mathbb{N}}$ and the locally complete $(DF)$-spaces $E$ such that for each $(e_{n})_{n \in \mathbb{N}} \in E^\mathbb{N}$ there exists an $E$-valued function $\mathbf{f}$ on $(0,\infty)$ (satisfying a mild regularity condition) such that $$\int_{0}^{\infty} t^{z_{n}} \mathbf{f}(t) dt = e_{n}, \qquad \forall n \in \mathbb{N},$$ where the integral should be understood as a Pettis integral. Moreover, in this case, we show that there always exists a solution $\mathbf{f}$ that is smooth on $(0,\infty)$ and satisfies certain optimal growth bounds near $0$ and $\infty$. The scalar-valued case $(E = \mathbb{C})$ was treated by Durán [Math. Nachr. 158 (1992), 175-194]. Our work is based upon his result.

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arXiv2024

An extension result for $(LB)$-spaces and the surjectivity of tensorized mappings

We study an extension problem for continuous linear maps in the setting of $(LB)$-spaces. More precisely, we characterize the pairs $(E,Z)$, where $E$ is a locally complete space with a fundamental sequence of bounded sets and $Z$ is an $(LB)$-space, such that for every exact sequence of $(LB)$-spaces $$ 0 \rightarrow X \xrightarrowι Y \rightarrow Z \rightarrow 0$$ the map $$ L(Y,E) \to L(X, E), ~ T \mapsto T \circ ι$$ is surjective, meaning that each continuous linear map $X \to E$ can be extended to a continuous linear map $Y \to E$ via $ι$, under some mild conditions on $E$ or $Z$ (e.g. one of them is nuclear). We use our extension result to obtain sufficient conditions for the surjectivity of tensorized maps between Fréchet-Schwartz spaces. As an application of the latter, we study vector-valued Eidelheit type problems. Our work is inspired by and extends results of Vogt [24].

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arXiv2023

Kernel theorems for Beurling-Björck type spaces

We prove new kernel theorems for a general class of Beurling-Björck type spaces. In particular, our results cover the classical Beurling-Björck spaces $\mathcal{S}^{(ω)}_{(η)}$ and $\mathcal{S}^{\{ω\}}_{\{η\}}$ defined via weight functions $ω$ and $η$.

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arXiv2022

Sequence space representations for translation-modulation invariant function and distribution spaces

We provide sequence space representations for the test function space $\mathcal{D}_{E}$ and the distribution space $\mathcal{D}^{\prime}_{E}$ associated to a Banach space $E$ belonging to a broad class of translation-modulation invariant Banach spaces of distributions. The spaces $\mathcal{D}_{E}$ and $\mathcal{D}^{\prime}_{E}$ generalize the classical Schwartz spaces $\mathcal{D}_{L^p}$ and $\mathcal{D}^{\prime}_{L^p}$, respectively. Our proof is based on Gabor frame characterizations of $\mathcal{D}_{E}$ and $\mathcal{D}^{\prime}_{E}$, which are also established here and are of independent interest. We recover in a unified way some known sequence space representations as well as obtain several new ones.

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arXiv2022

The global solvability of the Kirchhoff equation with Sobolev data

We consider linear and non-linear Cauchy equations in the context of Sobolev spaces. In particular, we show the global existence of solutions to the Kirchhoff equation with initial data in the Sobolev spaces, a problem that has been open for more than eighty years. Our proof is based on a new uniform estimate for solutions to the linear equation with time-dependent coefficient and a fixed point argument. As an immediate consequence of our result, the global solvability of the Kirchhoff equation with initial data in the Gevrey spaces is also obtained.

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arXiv2022

A note on the barrelledness of weighted $(PLB)$-spaces of ultradifferentiable functions

In this note we consider weighted $(PLB)$-spaces of ultradifferentiable functions defined via a weight function and a weight system, as introduced in our previous work [4]. We provide a complete characterization of when these spaces are ultrabornological and barrelled in terms of the defining weight system, thereby improving the main Theorem 5.1 of [4]. In particular, we obtain that the multiplier space of the Gelfand-Shilov space $Σ^{r}_{s}(\mathbb{R}^{d})$ of Beurling type is ultrabornological, whereas the one of the Gelfand-Shilov space $\mathcal{S}^{r}_{s}(\mathbb{R}^{d})$ of Roumieu type is not barrelled.

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arXiv2022

A note on composition operators between weighted spaces of smooth functions

For certain weighted locally convex spaces $X$ and $Y$ of one real variable smooth functions, we characterize the smooth functions $φ: \mathbb{R} \to \mathbb{R}$ for which the composition operator $C_φ: X \to Y, \, f \mapsto f \circ φ$ is well-defined and continuous. This problem has been recently considered for $X = Y$ being the space $\mathscr{S}$ of rapidly decreasing smooth functions [1] and the space $\mathscr{O}_M$ of slowly increasing smooth functions [2]. In particular, we recover both these results as well as obtain a characterization for $X =Y$ being the space $\mathscr{O}_C$ of very slowly increasing smooth functions.

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arXiv2022

Weighted $(PLB)$-spaces of ultradifferentiable functions and multiplier spaces

We study weighted $(PLB)$-spaces of ultradifferentiable functions defined via a weight function (in the sense of Braun, Meise and Taylor) and a weight system. We characterize when such spaces are ultrabornological in terms of the defining weight system. This generalizes Grothendieck's classical result that the space $\mathcal{O}_M$ of slowly increasing smooth functions is ultrabornological to the context of ultradifferentiable functions. Furthermore, we determine the multiplier spaces of Gelfand-Shilov spaces and, by using the above result, characterize when such spaces are ultrabornological. In particular, we show that the multiplier space of the space of Fourier ultrahyperfunctions is ultrabornological, whereas the one of the space of Fourier hyperfunctions is not.

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