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Jasson Vindas

Publications and source records attributed to Jasson Vindas.

At least 55 records · Page 3Linked to original sources
arXiv2020

Multiresolution expansions and wavelets in Gelfand-Shilov spaces

We study approximation properties generated by highly regular scaling functions and orthonormal wavelets. These properties are conveniently described in the framework of Gelfand-Shilov spaces. Important examples of multiresolution analyses for which our results apply arise in particular from Dziubański-Hernández construction of band-limited wavelets with subexponential decay. Our results are twofold. Firstly, we obtain approximation properties of multiresolution expansions of Gelfand-Shilov functions and (ultra)distributions. Secondly, we establish convergence of wavelet series expansions in the same regularity framework.

math.FA↗
arXiv2019

On the projective description of spaces of ultradifferentiable functions of Roumieu type

We provide a projective description of the space $\mathcal{E}^{\{\mathfrak{M}\}}(Ω)$ of ultradifferentiable functions of Roumieu type, where $Ω$ 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.

math.FA↗
arXiv2019

On the space of ultradistributions vanishing at infinity

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

math.FA↗
arXiv2019

A non-linear theory of infrahyperfunctions

We develop a nonlinear theory for infrahyperfunctions (also referred to as quasianalytic (ultra)distributions by L. Hörmander). 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.

math.FA↗
arXiv2019

Tauberian class estimates for vector-valued distributions

We study Tauberian properties of regularizing transforms of vector-valued tempered distributions, that is, transforms of the form $M^{\mathbf{f}}_φ(x,y)=(\mathbf{f}\astφ_{y})(x)$, where the kernel $φ$ is a test function and $φ_{y}(\cdot)=y^{-n}φ(\cdot/y)$. We investigate conditions which ensure that a distribution that a priori takes values in locally convex space actually takes values in a narrower Banach space. Our goal is to characterize spaces of Banach space valued tempered distributions in terms of so-called class estimates for the transform $M^{\mathbf{f}}_φ(x,y)$. Our results generalize and improve earlier Tauberian theorems of Drozhzhinov and Zav'yalov [Sb. Math. 194 (2003), 1599-1646]. Special attention is paid to find the optimal class of kernels $φ$ for which these Tauberian results hold.

math.FA↗
arXiv2019

On Diamond's $L^1$ criterion for asymptotic density of Beurling generalized integers

We give a short proof of the $L^{1}$ criterion for Beurling generalized integers to have a positive asymptotic density. We actually prove the existence of density under a weaker hypothesis. We also discuss related sufficient conditions for the estimate $m(x)=\sum_{n_{k}\leq x} μ(n_k)/n_k=o(1)$, with $μ$ the Beurling analog of the Moebius function.

math.NT↗
arXiv2018

Structural theorems for quasiasymptotics of ultradistributions

We provide complete structural theorems for the so-called quasiasymptotic behavior of non-quasianalytic ultradistributions. As an application of these results, we obtain descriptions of quasiasymptotic properties of regularizations at the origin of ultradistributions and discuss connections with Gelfand-Shilov type spaces.

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↗
arXiv2018

Note on the absence of remainders in the Wiener-Ikehara theorem

We show that it is impossible to get a better remainder than the classical one in the Wiener-Ikehara theorem even if one assumes analytic continuation of the Mellin transform after subtraction of the pole to a half-plane. We also prove a similar result for the Ingham-Karamata theorem.

math.CA↗
arXiv2018

Translation-modulation invariant Banach spaces of ultradistributions

We introduce and study a new class of translation-modulation invariant Banach spaces of ultradistributions. These spaces show stability under Fourier transform and tensor products; furthermore, they have a natural Banach convolution module structure over a certain associated Beurling algebra, as well as a Banach multiplication module structure over an associated Wiener-Beurling algebra. We also investigate a new class of modulation spaces, the Banach spaces of ultradistributions $\mathcal{M}^F$ on $\mathbb{R}^{d}$, associated to translation-modulation invariant Banach spaces of ultradistributions $F$ on $\mathbb{R}^{2d}$.

math.FA↗
arXiv2018

Optimal Tauberian constant in Ingham's theorem for Laplace transforms

It is well known that there is an absolute constant $\mathfrak{C}>0$ such that if the Laplace transform $G(s)=\int_{0}^{\infty}ρ(x)e^{-s x}\:\mathrm{d}x$ of a bounded function $ρ$ has analytic continuation through every point of the segment $(-iλ,iλ)$ of the imaginary axis, then $$ \limsup_{x\to\infty} \left|\int_{0}^{x}ρ(u)\:\mathrm{d}u - G(0)\right|\leq \frac{ \mathfrak{C}}λ \: \limsup_{x\to\infty} |ρ(x)|. $$ The best known value of the constant $\mathfrak{C}$ was so far $\mathfrak{C}=2$. In this article we show that the inequality holds with $\mathfrak{C}=π/2$ and that this value is best possible. We also sharpen Tauberian constants in finite forms of other related complex Tauberian theorems for Laplace transforms.

math.CA↗
arXiv2018

Spectral asymptotics for infinite order pseudo-differential operators

We study spectral properties of a class of global infinite order pseudo-differential operators and obtain the asymptotic behaviour of the spectral counting functions of such operators. Unlike their finite order counterparts, their spectral asymptotics are not of power-log-type but of log-type. The ultradistributional setting of such operators of infinite order makes the theory more complex so that the standard finite order global Weyl calculus cannot be used in this context.

math.SP↗
arXiv2018

New classes of weighted Hölder-Zygmund spaces and the wavelet transform

We provide a new and elementary proof of the continuity theorem for the wavelet and left-inverse wavelet transforms on the spaces $ \mathcal{S}_0(\mathbb{R}^n) $ and $ \mathcal{S}(\mathbb{H}^{n+1})$. We then introduce and study a new class of weighted Hölder-Zygmund spaces, where the weights are regularly varying functions. The analysis of these spaces is carried out via the wavelet transform and generalized Littlewood-Paley pairs.

math.FA↗
arXiv2017

On quasianalytic classes of Gelfand-Shilov type. Parametrix and convolution

We develop a convolution theory for quasianalytic ultradistributions of Gelfand-Shilov type. We also construct a special class of ultrapolynomials, and use it as a base for the parametrix method in the study of new topological and structural properties of several quasianalytic spaces of functions and ultradistributions. In particular, our results apply to Fourier hyperfunctions and Fourier ultra-hyperfunctions.

math.FA↗
arXiv2017

Ultradistributional boundary values of harmonic functions on the sphere

We present a theory of ultradistributional boundary values for harmonic functions defined on the Euclidean unit ball. We also give a characterization of ultradifferentiable functions and ultradistributions on the sphere in terms of their spherical harmonic expansions. To this end, we obtain explicit estimates for partial derivatives of spherical harmonics, which are of independent interest and refine earlier estimates by Calderon and Zygmund. We apply our results to characterize the support of ultradistributions on the sphere via Abel summability of their spherical harmonic expansions.

math.FA↗
arXiv2017

General Stieltjes moment problems for rapidly decreasing smooth functions

We give (necessary and sufficient) conditions over a sequence $\left\{ f_{n}\right\} _{n=0}^{\infty}$ of functions under which every generalized Stieltjes moment problem \[ \int_{0}^{\infty} f_{n}(x)ϕ(x)\mathrm{d} x=a_{n}, \ \ \ n\in\mathbb{N}, \] has solutions $ϕ\in\mathcal{S}(\mathbb{R})$ with $\operatorname*{supp} ϕ\subseteq[0,\infty)$. Furthermore, we consider more general problems of this kind for measure or distribution sequences $\left\{ f_{n}\right\} _{n=0}^{\infty}$. We also study vector moment problems with values in Frechet spaces and multidimensional moment problems.

math.FA↗