arXiv subjects

Gerhard Schindl

Publications and source records attributed to Gerhard Schindl.

At least 55 records · Page 3Linked to original sources

arXiv2020

Nuclear global spaces of ultradifferentiable functions in the matrix weighted setting

We prove that the Hermite functions are an absolute Schauder basis for many global weighted spaces of ultradifferentiable functions in the matrix weighted setting and we determine also the corresponding coefficient spaces, thus extending previous work by Langenbruch. As a consequence we give very general conditions for these spaces to be nuclear. In particular, we obtain the corresponding results for spaces defined by weight functions.

math.FA↗

arXiv2019

On the construction of large Algebras not contained in the image of the Borel map

The Borel map $j^{\infty}$ takes germs at 0 of smooth functions to the sequence of iterated partial derivatives at 0. It is well known that the restriction of $j^{\infty}$ to the germs of quasianalytic ultradifferentiable classes which are strictly containing the real analytic functions can never be onto the corresponding sequence space. In a recent paper the authors have studied the size of the image of $j^{\infty}$ by using different approaches and worked in the general setting of quasianalytic ultradifferentiable classes defined by weight matrices. The aim of this paper is to show that the image of $j^{\infty}$ is also small with respect to the notion of algebrability and we treat both the Cauchy product (convolution) and the pointwise product. In particular, a deep study of the stability of the considered spaces under the pointwise product is developed.

math.FA↗

arXiv2019

The surjectivity of the Borel mapping in the mixed setting for ultradifferentiable ramification spaces

We consider r-ramification ultradifferentiable classes, introduced by J. Schmets and M. Valdivia in order to study the surjectivity of the Borel map, and later on also exploited by the authors in the ultraholomorphic context. We characterize quasianalyticity in such classes, extend the results of Schmets and Valdivia about the image of the Borel map in a mixed ultradifferentiable setting, and obtain a version of the Whitney extension theorem in this framework.

math.FA↗

arXiv2019

Almost analytic extensions of ultradifferentiable functions with applications to microlocal analysis

We review and extend the description of ultradifferentiable functions by their almost analytic extensions, i.e., extensions to the complex domain with specific vanishing rate of the $\bar \partial$-derivative near the real domain. We work in a general uniform framework which comprises the main classical ultradifferentiable classes but also allows to treat unions and intersections of such. The second part of the paper is devoted to applications in microlocal analysis. The ultradifferentiable wave front set is defined in this general setting and characterized in terms of almost analytic extensions and of the FBI transform. This allows to extend its definition to ultradifferentiable manifolds. We also discuss ultradifferentiable versions of the elliptic regularity theorem and obtain a general quasianalytic Holmgren uniqueness theorem.

math.AP↗

arXiv2019

Ultraholomorphic extension theorems in the mixed setting

The aim of this work is to generalize the ultraholomorphic extension theorems from V. Thilliez in the weight sequence setting and from the authors in the weight function setting (of Roumieu type) to a mixed framework. Such mixed results have already been known for ultradifferentiable classes and it seems natural that they have ultraholomorphic counterparts. In order to have control on the opening of the sectors in the Riemann surface of the logarithm for which the extension theorems are valid we are introducing new mixed growth indices which are generalizing the known ones for weight sequences and functions. As it turns out, for the validity of mixed extension results the so-called order of quasianalyticity (introduced by the second author for weight sequences) is becoming important.

math.CV↗

arXiv2019

Nuclearity of rapidly decreasing ultradifferentiable functions and time-frequency analysis

We use techniques from time-frequency analysis to show that the space $\mathcal S_ω$ of rapidly decreasing $ω$-ultradifferentiable functions is nuclear for every weight function $ω(t)=o(t)$ as $t$ tends to infinity. Moreover, we prove that, for a sequence $(M_p)_p$ satisfying the classical condition $(M1)$ of Komatsu, the space of Beurling type $\mathcal S_{(M_p)}$ when defined with $L^{2}\,$norms is nuclear exactly when condition $(M2)'$ of Komatsu holds.

math.FA↗

arXiv2018

On the extension of Whitney ultrajets, II

We characterize the validity of the Whitney extension theorem in the ultradifferentiable Roumieu setting with controlled loss of regularity. Specifically, we show that in the main Theorem 1.3 of [15] condition (1.3) can be dropped. Moreover, we clarify some questions that remained open in [15].

math.CA↗

arXiv2018

How far is the Borel map from being surjective in quasianalytic ultradifferentiable classes?

The Borel map takes germs at 0 of smooth functions to the sequence of iterated partial derivatives at 0. In the literature, it is well known that the restriction of this mapping to the germs of quasianalytic ultradifferentiable classes which are strictly containing the real analytic functions can never be onto the corresponding sequence space. In this paper, we are interested in studying how large the image of the Borel map is and we investigate the size and the structure of this image by using different approaches (Baire residuality, prevalence and lineability). We give an answer to this question in the very general setting of quasianalytic ultradifferentiable classes defined by weight matrices, which contains as particular cases the classes defined by a single weight sequence or by a weight function.

math.FA↗

arXiv2018

Indices of O-regular variation for weight functions and weight sequences

A plethora of spaces in Functional Analysis (Braun-Meise-Taylor and Carleman ultradifferentiable and ultraholomorphic classes; Orlicz, Besov, Lipschitz, Lebesque spaces, to cite the main ones) are defined by means of a weighted structure, obtained from a weight function or sequence subject to standard conditions entailing desirable properties (algebraic closure, stability under operators, interpolation, etc.) for the corresponding spaces. The aim of this paper is to stress or reveal the true nature of these diverse conditions imposed on weights, appearing in a scattered and disconnected way in the literature: they turn out to fall into the framework of O-regular variation, and many of them are equivalent formulations of one and the same feature. Moreover, we study several indices of regularity/growth for both functions and sequences, which allow for the rephrasing of qualitative properties in terms of quantitative statements.

math.FA↗

arXiv2018

Sectorial extensions for ultraholomorphic classes defined by weight functions

We prove an extension theorem for ultraholomorphic classes defined by so-called Braun-Meise-Taylor weight functions and transfer the proofs from the single weight sequence case from V. Thilliez [28] to the weight function setting. We are following a different approach than the results obtained in [11], more precisely we are working with real methods by applying the ultradifferentiable Whitney-extension theorem. We are treating both the Roumieu and the Beurling case, the latter one is obtained by a reduction from the Roumieu case.

math.FA↗

arXiv2018

Injectivity and surjectivity of the asymptotic Borel map in Carleman ultraholomorphic classes

We study the injectivity and surjectivity of the Borel map in three instances: in Roumieu-Carleman ultraholomorphic classes in unbounded sectors of the Riemann surface of the logarithm, and in classes of functions admitting, uniform or nonuniform, asymptotic expansion at the corresponding vertex. These classes are defined in terms of a log-convex sequence $\mathbb{M}$ of positive real numbers. Injectivity had been solved in two of these cases by S. Mandelbrojt and B. Rodríguez-Salinas, respectively, and we completely solve the third one by means of the theory of proximate orders. A growth index $ω(\mathbb{M})$ turns out to put apart the values of the opening of the sector for which injectivity holds or not. In the case of surjectivity, only some partial results were available by J. Schmets and M. Valdivia and by V. Thilliez, and this last author introduced an index $γ(\mathbb{M})$ (generally different from $ω(\mathbb{M})$) for this problem, whose optimality was not established except for the Gevrey case. We considerably extend here their results, proving that $γ(\mathbb{M})$ is indeed optimal in some standard situations (for example, as far as $\mathbb{M}$ is strongly regular) and puts apart the values of the opening of the sector for which surjectivity holds or not.

math.CV↗

arXiv2018

Log-convex sequences and nonzero proximate orders

Summability methods for ultraholomorphic classes in sectors, defined in terms of a strongly regular sequence $\mathbb{M}=(M_p)_{p\in\mathbb{N}_0}$, have been put forward by A. Lastra, S. Malek and the second author [1], and their validity depends on the possibility of associating to $\mathbb{M}$ a nonzero proximate order. We provide several characterizations of this and other related properties, in which the concept of regular variation for functions and sequences plays a prominent role. In particular, we show how to construct well-behaved strongly regular sequences from nonzero proximate orders. [1] A. Lastra, S. Malek and J. Sanz, Summability in general Carleman ultraholomorphic classes, J. Math. Anal. Appl. 430 (2015), 1175--1206.

math.CA↗

arXiv2018

Sectorial extensions, via Laplace transforms, in ultraholomorphic classes defined by weight functions

We prove several extension theorems for Roumieu ultraholomorphic classes of functions in sectors of the Riemann surface of the logarithm which are defined by means of a weight function or weight matrix. Our main aim is to transfer the results of V. Thilliez from the weight sequence case to these different, or more general, frameworks. The technique rests on the construction of suitable kernels for a truncated Laplace-like integral transform, which provides the solution without resorting to Whitney-type extension results for ultradifferentiable classes. As a byproduct, we obtain an extension in a mixed weight-sequence setting in which assumptions on the sequence are minimal.

math.FA↗

arXiv2017

On the extension of Whitney ultrajets

We prove necessary and sufficient conditions for the validity of Whitney's extension theorem in the ultradifferentiable Roumieu setting with controlled loss of regularity.

math.CA↗

arXiv2017

On the Borel mapping in the quasianalytic setting

The Borel mapping takes germs at $0$ of smooth functions to the sequence of iterated partial derivatives at $0$. We prove that the Borel mapping restricted to the germs of any quasianalytic ultradifferentiable class strictly larger than the real analytic class is never onto the corresponding sequence space.

math.CA↗

arXiv2017

A Phragmén-Lindelöf theorem via proximate orders, and the propagation of asymptotics

We prove that, for asymptotically bounded holomorphic functions in a sector in $\mathbb{C}$, an asymptotic expansion in a single direction towards the vertex with constraints in terms of a logarithmically convex sequence admitting a nonzero proximate order entails asymptotic expansion in the whole sector with control in terms of the same sequence. This generalizes a result by A. Fruchard and C. Zhang for Gevrey asymptotic expansions, and the proof strongly rests on a suitably refined version of the classical Phragmén-Lindelöf theorem, here obtained for functions whose growth in a sector is specified by a nonzero proximate order in the sense of E. Lindelöf and G. Valiron.

math.CV↗

arXiv2017

Extension of Whitney jets of controlled growth

We revisit Whitney's extension theorem in the ultradifferentiable Roumieu setting. Based on the description of ultradifferentiable classes by weight matrices, we extend results on how growth constraints on Whitney jets on arbitrary compact subsets in $\mathbb R^n$ are preserved by their extensions to $\mathbb R^n$. More precisely, for any admissible class $\mathcal C$ of ultradifferentiable functions on $\mathbb R^n$ we determine a class $\mathcal C'$ such that all ultradifferentiable Whitney jets of class $\mathcal C'$ on arbitrary compact subsets admit extensions in $\mathcal C$. The class $\mathcal C'$ can be explicitly computed from $\mathcal C$.

math.CA↗

arXiv2016

Characterization of ultradifferentiable test functions defined by weight matrices in terms of their Fourier transform

We prove that functions with compact support in non-quasianalytic classes of Roumieu-type and of Beurling-type defined by a weight matrix with some mild regularity conditions can be characterized by the decay properties of their Fourier transform. For this we introduce the abstract technique of constructing from the original matrix multi-index matrices and associated function spaces. We study the behaviour of this construction in detail and characterize its stability. Moreover non-quasianalyticity of the classes is characterized.

math.FA↗