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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-Bj\"orck spaces $\mathcal{S}^{[\omega]}_{[\eta]}$ (defined via weight functions $\omega$ and $\eta$)., Comment: 15 pages

We prove that the spaces of smooth and ultradifferentiable vectors associated with a representation of a real Lie group on a Fr\'{e}chet space $E$ are quasinormable if $E$ is so. A similar result is shown to hold for the linear topological invariant $(\Omega)$. In the ultradifferentiable case, our results particularly apply to spaces of Gevrey vectors of Beurling type. As an application, we study the quasinormability and the property $(\Omega)$ for a broad class of Fr\'{e}chet spaces of smooth and ultradifferentiable functions on Lie groups globally defined via families of weight functions., Comment: 33 pages

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 Dur\'an [Math. Nachr. 158 (1992), 175-194]. Our work is based upon his result., Comment: 13 pages

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 $$ 0 \rightarrow X \xrightarrow{\iota} Y \rightarrow Z \rightarrow 0$$ the map $$ L(Y,E) \to L(X, E), ~ T \mapsto T \circ \iota $$ is surjective, meaning that each continuous linear map $X \to E$ can be extended to a continuous linear map $Y \to E$ via $\iota$, 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 Fr\'{e}chet-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]., Comment: 30 pages

We characterize the condition $(\Omega)$ for smooth kernels of partial differential operators in terms of the existence of shifted fundamental solutions satisfying certain properties. The conditions $(P\Omega)$ and $(P\overline{\overline{\Omega}})$ for distributional kernels are characterized in a similar way. By lifting theorems for Fr\'echet spaces and (PLS)-spaces, this provides characterizations of the problem of parameter dependence for smooth and distributional solutions of differential equations by shifted fundamental solutions. As an application, we give a new proof of the fact that the space $\{ f \in \mathscr{E}(X) \, | \, P(D)f = 0\}$ satisfies $(\Omega)$ for any differential operator $P(D)$ and any open convex set $X \subseteq \mathbb{R}^d$., Comment: 15 pages

We prove quantitative Runge type approximation results for spaces of smooth zero solutions of several classes of linear partial differential operators with constant coefficients. Among others, we establish such results for arbitrary operators on convex sets, elliptic operators, parabolic operators, and the wave operator in one spatial variable. Our methods are inspired by the study of linear topological invariants for kernels of partial differential operators. As a part of our work, we also show a qualitative Runge type approximation theorem for subspace elliptic operators, which seems to be new and of independent interest., Comment: 27 pages; comments welcome; v2: fixed some typos; reference to implicit approximation results as well as a global approximation result due to Malgrange has been added in the introduction which allows to choose approximating global solutions as linear combinations of exponential solutions; slight refomulation of Theorems 1.1, 1.2 and 1.3; Remark 5.10 and Example 5.11 have been removed

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 $\Sigma^{r}_{s}(\mathbb{R}^{d})$ 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., Comment: 13 pages

We establish the condition $(\Omega)$ for smooth kernels of various types of convolution and differential operators. By the $(DN)$-$(\Omega)$ splitting theorem of Vogt and Wagner, this implies that these operators are surjective on the corresponding spaces of vector-valued smooth functions with values in a product of Montel $(DF)$-spaces whose strong duals satisfy the condition $(DN)$, e.g., the space $\mathscr{D}'(Y)$ of distributions over an open set $Y \subseteq \mathbb{R}^n$ or the space $\mathscr{S}'(\mathbb{R}^n)$ of tempered distributions. Most notably, we show that: $(i)$ $\mathscr{E}_P(X) = \{ f \in \mathscr{E}(X) \, | \, P(D)f = 0 \}$ satisfies $(\Omega)$ for any differential operator $P(D)$ and any open convex set $X \subseteq \mathbb{R}^d$. $(ii)$ Let $P\in\mathbb{C}[\xi_1,\xi_2]$ and $X \subseteq \mathbb{R}^2$ open be such that $P(D):\mathscr{E}(X)\rightarrow\mathscr{E}(X)$ is surjective. Then, $\mathscr{E}_P(X)$ satisfies $(\Omega)$. $(iii)$ Let $\mu \in \mathscr{E}'(\mathbb{R}^d)$ be such that $ \mathscr{E}(\mathbb{R}^d) \rightarrow \mathscr{E}(\mathbb{R}^d), \, f \mapsto \mu \ast f$ is surjective. Then, $ \{ f \in \mathscr{E}(\mathbb{R}^d) \, | \, \mu \ast f = 0 \}$ satisfies $(\Omega)$. The central result in this paper states that the space of smooth zero solutions of a general convolution equation satisfies the condition $(\Omega)$ if and only if the space of distributional zero solutions of the equation satisfies the condition $(P\Omega)$. The above and related results then follow from known results concerning $(P\Omega)$ for distributional kernels of convolution and differential operators., Comment: 17 pages; correction of typos; accepted for publication in Journal of Functional Analysis

For certain weighted locally convex spaces $X$ and $Y$ of one real variable smooth functions, we characterize the smooth functions $\varphi: \mathbb{R} \to \mathbb{R}$ for which the composition operator $C_\varphi: X \to Y, \, f \mapsto f \circ \varphi$ 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., Comment: 9 pages

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., Comment: 17 pages

We provide explicit commutative sequence space representations for classical function and distribution spaces on the real half-line. This is done by evaluating at the Fourier transforms of the elements of an orthonormal wavelet basis., Comment: 8 pages

We provide explicit sequence space representations for the test function and distribution spaces occurring in the Valdivia-Vogt structure tables by making use of Wilson bases generated by compactly supported smooth windows. Furthermore, we show that these kind of bases are common unconditional Schauder bases for all separable spaces occurring in these tables. Our work implies that the Valdivia-Vogt structure tables for test functions and distributions may be interpreted as one large commutative diagram., Comment: 14 pages

We obtain Gabor frame characterisations of modulation spaces defined via a class of translation-modulation invariant Banach spaces of distributions that was recently introduced in $[10]$. We show that these spaces admit an atomic decomposition through Gabor expansions and that they are characterised by summability properties of their Gabor coefficients. Furthermore, we construct a large space of admissible windows. This generalises several fundamental results for the classical modulation spaces $ M^{p,q}_{w}$. Due to the absence of solidity assumptions on the Banach spaces defining these modulation spaces, the methods used for the spaces $M^{p,q}_{w}$ (or, more generally, in coorbit space theory) fail in our setting and we develop here a new approach based on the twisted convolution., Comment: 42 pages

We study boundary values of harmonic functions in spaces of quasianalytic functionals and spaces of ultradistributions of non-quasianalytic type. As an application, we provide a new approach to H\"ormander's support theorem for quasianalytic functionals. Our main technical tool is a description of ultradifferentiable functions by almost harmonic functions, a concept that we introduce in this article. We work in the setting of ultradifferentiable classes defined via weight matrices. In particular, our results simultaneously apply to the two standard classes defined via weight sequences and via weight functions., Comment: 25 pages

We study ultradistributional boundary values of zero solutions of a hypoelliptic constant coefficient partial differential operator $P(D) = P(D_x, D_t)$ on $\mathbb{R}^{d+1}$. Our work unifies and considerably extends various classical results of Komatsu and Matsuzawa about boundary values of holomorphic functions, harmonic functions and zero solutions of the heat equation in ultradistribution spaces. We also give new proofs of several results of Langenbruch [23] about distributional boundary values of zero solutions of $P(D)$., Comment: 31 pages

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., Comment: 26 pages

We give a complete solution to the Borel-Ritt problem in non-uniform spaces $\mathscr{A}^-_{(M)}(S)$ of ultraholomorphic functions of Beurling type, where $S$ is an unbounded sector of the Riemann surface of the logarithm and $M$ is a strongly regular weight sequence. Namely, we characterize the surjectivity and the existence of a continuous linear right inverse of the asymptotic Borel map on $\mathscr{A}^-_{(M)}(S)$ in terms of the aperture of the sector $S$ and the weight sequence $M$. Our work improves previous results by Thilliez [10] and Schmets and Valdivia [9]., Comment: 13 pages; updated version

For two types of moderate growth representations of $(\mathbb{R}^d,+)$ on sequentially complete locally convex Hausdorff spaces (including F-representations [J. Funct. Anal. 262 (2012), 667-681], we introduce Denjoy-Carleman classes of ultradifferentiable vectors and show a strong factorization theorem of Dixmier-Malliavin type for them. In particular, our factorization theorem solves [Conjecture 6.; J. Funct. Anal. 262 (2012), 667-681] for analytic vectors of representations of $G =(\mathbb{R}^d,+)$. As an application, we show that various convolution algebras and modules of ultradifferentiable functions satisfy the strong factorization property., Comment: 27 pages

We obtain sequence space representations for a class of Fr\'echet spaces of entire functions with rapid decay on horizontal strips. In particular, we show that the projective Gelfand-Shilov spaces $\Sigma^1_\nu$ and $\Sigma^\nu_1$ are isomorphic to $\Lambda_{\infty}(n^{1/(\nu+1)})$ for $\nu > 0$., Comment: 19pages; updated version

We provide a projective description of the space $\mathcal{E}^{\{\mathfrak{M}\}}(\Omega)$ of ultradifferentiable functions of Roumieu type, where $\Omega$ is an arbitrary open set in $\mathbb{R}^d$ and $\mathfrak{M}$ is a weight matrix satisfying the analogue of Komatsu's condition $(M.2)'$. In particular, we obtain in a unified way projective descriptions of ultradifferentiable classes defined via a single weight sequence (Denjoy-Carleman approach) and via a weight function (Braun-Meise-Taylor approach) under considerably weaker assumptions than in earlier versions of these results., Comment: 8 pages

We study the nuclearity of the Gelfand-Shilov spaces $\mathcal{S}^{(\mathfrak{M})}_{(\mathscr{W})}$ and $\mathcal{S}^{\{\mathfrak{M}\}}_{\{\mathscr{W}\}}$, defined via a weight (multi-)sequence system $\mathfrak{M}$ and a weight function system $\mathscr{W}$. We obtain characterizations of nuclearity for these function spaces that are counterparts of those for K\"{o}the sequence spaces. As an application, we prove new kernel theorems. Our general framework allows for a unified treatment of the Gelfand-Shilov spaces $\mathcal{S}^{(M)}_{(A)}$ and $\mathcal{S}^{\{M\}}_{\{A\}}$ (defined via weight sequences $M$ and $A$) and the Beurling-Bj\"orck spaces $\mathcal{S}^{(\omega)}_{(\eta)}$ and $\mathcal{S}^{\{\omega\}}_{\{\eta\}}$ (defined via weight functions $\omega$ and $\eta$). Our results cover anisotropic cases as well., Comment: 24 pages

We show that the space $\mathcal{S}'(\Gamma)$ of Laplace transformable distributions, where $\Gamma \subseteq \mathbb{R}^d$ is a non-empty convex open set, is an ultrabornological (PLS)-space. Moreover, we determine an explicit topological predual of $\mathcal{S}'(\Gamma)$., Comment: 10 pages

We characterize the nuclearity of the Beurling-Bj\"{o}rck spaces $\mathcal{S}^{(\omega)}_{(\eta)}(\mathbb{R}^d)$ and $\mathcal{S}^{\{\omega\}}_{\{\eta\}}(\mathbb{R}^d)$ in terms of the defining weight functions $\omega$ and $\eta$., Comment: 10 pages

We study the structural and linear topological properties of the space $\dot{\mathcal{B}}^{\prime \ast}_{\omega}$ of ultradistributions vanishing at infinity (with respect to a weight function $\omega$). Particularly, we show the first structure theorem for $\dot{\mathcal{B}}^{\prime \ast}_{\omega}$ under weaker hypotheses than were known so far. As an application, we determine the structure of the S-asymptotic behavior of ultradistributions., Comment: 18 pages

We characterize the surjectivity and the existence of a continuous linear right inverse of the Stieltjes moment mapping on Gelfand-Shilov spaces, both of Beurling and Roumieu type, in terms of their defining weight sequence. As a corollary, we obtain some new results about the Borel-Ritt problem in spaces of ultraholomorphic functions on the upper half-plane., Comment: 25 pages

We show that the spaces $\mathcal{E}_{\{\omega\}}(\Omega)$ of ultradifferentiable functions of Roumieu type satisfy the dual interpolation estimate for small theta, where $\omega$ is a quasianalytic weight function and $\Omega$ is an arbitrary open subset of $\mathbb{R}^d$. This result was previously shown by Bonet and Doma\'nski [2] under the additional assumptions that $\Omega$ is convex and $\omega$ satisfies the condition $(\alpha_1)$. In particular, our work solves Problem 9.7 in [1]., Comment: 20 pages

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., Comment: 18 pages

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 Fr\'echet 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., Comment: 14 pages

We discuss the structural and topological properties of a general class of weighted $L^1$ convolutor spaces. Our theory simultaneously applies to weighted $\mathcal{D}'_{L^1}$ spaces as well as to convolutor spaces of the Gelfand-Shilov spaces $\mathcal{K}\{M_p\}$. In particular, we characterize the sequences of weight functions $(M_p)_{p \in \mathbb{N}}$ for which the space of convolutors of $\mathcal{K}\{M_p\}$ is ultrabornological, thereby generalizing Grothendieck's classical result for the space $\mathcal{O}'_{C}$ of rapidly decreasing distributions. Our methods lead to the first direct proof of the completeness of the space $\mathcal{O}_{C}$ of very slowly increasing smooth functions., Comment: 34 pages

We provide a projective description for a class of generalized Gelfand-Shilov spaces of Roumieu type. In particular, our results apply to the classical Gelfand-Shilov spaces and weighted $L^\infty$-spaces of ultradifferentiable functions of Roumieu type., Comment: 10 pages

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., Comment: 35 pages

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.

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., Comment: 35 pages

We develop a nonlinear theory for infrahyperfunctions (also referred to as quasianalytic (ultra)distributions by L. H\"{o}rmander). In the hyperfunction case our work can be summarized as follows. We construct a differential algebra that contains the space of hyperfunctions as a linear differential subspace and in which the multiplication of real analytic functions coincides with their ordinary product. Moreover, by proving an analogue of Schwartz's impossibility result for hyperfunctions, we show that this embedding is optimal. Our results fully solve an earlier question raised by M. Oberguggenberger., Comment: 24 pages

We introduce vector-valued Eidelheit sequences and obtain a characterization that generalizes Eidelheit's classical theorem (Studia Math. 6 (1936), 139-148). As an application, we discuss criteria for non $B$-completeness of completed tensor products., Comment: 11 pages

We study spaces of vector-valued quasianalytic functions and solve the first Cousin problem in these spaces., Comment: 23 pages

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., Comment: 39 pages

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., Comment: 29 pages

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 $f\in\mathcal{D}'(\Omega)$. Let $\Lambda$ be a lattice in $\mathbb{R}^{d}$ and let $U$ be an open convex neighborhood of the origin such that $U\cap\Lambda^{*}=\{0\}$. The analytic wave front set $WF_{A}(f)$ coincides with the complement in $\Omega\times(\mathbb{R}^{d}\setminus\{0\})$ of the set of points $(x_0,\xi_0)$ for which there are an open neighborhood $V\subset \Omega\cap (x_0+U)$ of $x_0$, an open conic neighborhood $\Gamma$ of $\xi_0$, and a bounded sequence $(f_p)_{p \in \mathbb{N}}$ in $\mathcal{E}'(\Omega\cap (x_0+U))$ with $f_p= f$ on $V$ such that for some $h > 0$ \[ \sup_{\mu \in \Gamma \cap \Lambda} |\widehat{f_p} (\mu)| |\mu|^p \leq h^{p+1}p!\:, \qquad \forall p \in \mathbb{N}. \], Comment: 21 pages