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Stevan Pilipović

Publications and source records attributed to Stevan Pilipović.

At least 37 records · Page 2Linked to original sources
arXiv2021

Infinite order $Ψ$DOs: Composition with entire functions, new Shubin-Sobolev spaces, and index theorem

We study global regularity and spectral properties of power series of the Weyl quantisation $a^w$, where $a(x,ξ) $ is a classical elliptic Shubin polynomial. For a suitable entire function $P$, we associate two natural infinite order operators to $a^{w}$, $P(a^w)$ and $(P\circ a)^{w},$ and prove that these operators and their lower order perturbations are globally Gelfand-Shilov regular. They have spectra consisting of real isolated eigenvalues diverging to $\infty$ for which we find the asymptotic behaviour of their eigenvalue counting function. In the second part of the article, we introduce Shubin-Sobolev type spaces by means of $f$-$Γ^{*,\infty}_{A_p,ρ}$-elliptic symbols, where $f $ is a function of ultrapolynomial growth and $Γ^{*,\infty}_{A_p,ρ}$ is a class of symbols of infinite order studied in this and our previous papers. We study the regularity properties of these spaces, and show that the pseudo-differential operators under consideration are Fredholm operators on them. Their indices are independent on the order of the Shubin-Sobolev spaces; finally, we show that the index can be expressed via a Fedosov-Hörmander integral formula.

math.AP↗
arXiv2020

Modulation spaces associated to tensor products of amalgam spaces

We identify the modulation spaces associated to tensor products of amalgam spaces having a large class of Banach spaces as their local component. As consequences of the main results, we describe the modulation spaces associated to tensor products of various $L^p$ spaces.

math.FA↗
arXiv2020

Boundary values in ultradistribution spaces related to extended Gevrey regularity

Following the well-known theory of Beurling and Roumieu ultradistributions, we investigate new spaces of ultradistributions as dual spaces of test functions which correspond to associated functions of logarithmic-type growth at infinity. In the given framework we prove that boundary values of analytic functions with the corresponding logarithmic growth rate towards the real domain are ultradistributions. The essential condition for that purpose, condition $(M.2)$ in the classical ultradistribution theory, is replaced by the new one, $\widetilde{(M.2)}$. For that reason, new techniques were performed in the proofs. As an application, we discuss the corresponding wave front sets.

math.FA↗
arXiv2020

Directional short-time Fourier transform of ultradistributions

We define and analyse the $k$-directional short-time Fourier transform and its synthesis operator over Gelfand Shilov spaces $\mathcal S^α_β(\mathbb R^n)$ and $\mathcal S^α_β(\mathbb R^{k+n})$ respectively, and their duals. Also, we investigate directional regular sets and their complements - directional wave fronts, for elements of $\mathcal S^{\prime α}_α(\mathbb R^n)$.

math.FA↗
arXiv2020

The Shearlet Transform and Lizorkin Spaces

We prove a continuity result for the shearlet transform when restricted to the space of smooth and rapidly decreasing functions with all vanishing moments. We define the dual shearlet transform, called here the shearlet synthesis operator, and we prove its continuity on the space of smooth and rapidly decreasing functions over $\mathbb{R}^2\times\mathbb{R}\times\mathbb{R}^\times$. Then, we use these continuity results to extend the shearlet transform to the space of Lizorkin distributions, and we prove its consistency with the classical definition for test functions.

math.FA↗
arXiv2020

Continuity properties of the shearlet transform and the shearlet synthesis operator on the Lizorkin type spaces

We develop a distributional framework for the shearlet transform $\mathcal{S}_ψ\colon\mathcal{S}_0(\mathbb{R}^2)\to\mathcal{S}(\mathbb{S})$ and the shearlet synthesis operator $\mathcal{S}^t_ψ\colon\mathcal{S}(\mathbb{S})\to\mathcal{S}_0(\mathbb{R}^2)$, where $\mathcal{S}_0(\mathbb{R}^2)$ is the Lizorkin test function space and $\mathcal{S}(\mathbb{S})$ is the space of highly localized test functions on the standard shearlet group $\mathbb{S}$. These spaces and their duals $\mathcal{S}_0^\prime (\mathbb R^2),\, \mathcal{S}^\prime (\mathbb{S})$ are called Lizorkin type spaces of test functions and distributions. We analyze the continuity properties of these transforms when the admissible vector $ψ$ belongs to $\mathcal{S}_0(\mathbb{R}^2)$. Then, we define the shearlet transform and the shearlet synthesis operator of Lizorkin type distributions as transpose mappings of the shearlet synthesis operator and the shearlet transform, respectively. They yield continuous mappings from $\mathcal{S}_0^\prime (\mathbb R^2)$ to $\mathcal{S}^\prime (\mathbb{S})$ and from $\mathcal{S}^\prime (\mathbb S)$ to $\mathcal{S}_0^\prime (\mathbb{R}^2)$. Furthermore, we show the consistency of our definition with the shearlet transform defined by direct evaluation of a distribution on the shearlets. The same can be done for the shearlet synthesis operator. Finally, we give a reconstruction formula for Lizorkin type distributions, from which follows that the action of such generalized functions can be written as an absolutely convergent integral over the standard shearlet group.

math.FA↗
arXiv2020

Semi-Bloch periodic functions, semi-anti-periodic functions and applications

In this paper, we introduce the notions of semi-Bloch periodic functions and semi-anti-periodic functions. Stepanov semi-Bloch periodic functions and Stepanov semi-anti-periodic functions are considered, as well. We analyze the invariance of introduced classes under the actions of convolution products and briefly explain how one can use the obtained results in the qualitative analysis of solutions of abstract inhomogeneous integro-differential equations.

math.FA↗
arXiv2020

Note on vector valued Hardy spaces related to analytic functions having distributional boundary values

Analytic functions defined on a tube domain $T^{C}\subset \mathbb{C}^{n}$ and taking values in a Banach space $X$ which are known to have $X$-valued distributional boundary values are shown to be in the Hardy space $H^{p}(T^{C},X)$ if the boundary value is in the vector valued Lebesgue space $L^{p}(\mathbb{R}^{n},X)$, where $1\leq p \leq \infty$ and $C$ is a regular open convex cone. Poisson integral transform representations of elements of $H^{p}(T^{C}, X)$ are also obtained for certain classes of Banach spaces, including reflexive Banach spaces.

math.CV↗
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

Localization of Fréchet frames and expansion of generalized functions

Matrix type operators with the off-diagonal decay of polynomial or sub-exponential types are revisited with weaker assumptions concerning row or column estimates, still giving the continuity results for the frame type operators. Such results are extended from Banach to Fréchet spaces. Moreover, the localization of Fréchet frames is used for the frame expansions of tempered distributions and a class of Beurling ultradistributions.

math.FA↗
arXiv2019

Equivalence of Ellipticity and Fredholmness in the Weyl-Hörmander calculus

The main result is that the Fredholm property of a $Ψ$DO acting on Sobolev spaces in the Weyl-Hörmander calculus and the ellipticity are equivalent for geodesically temperate Hörmanders metrics whose associated Planck's functions vanish at infinity. Additionally, we prove that when the Hörmander metric is geodesically temperate, and consequently the calculus is spectrally invariant, the inverse $λ\mapsto b_λ\in S(1,g)$ of every $\mathcal{C}^N$, $0\leq N\leq \infty$, map $λ\mapsto a_λ\in S(1,g)$ comprised of invertible elements on $L^2$ is again of class $\mathcal{C}^N$.

math.AP↗
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

A Paley-Wiener theorem in extended Gevrey regularity

In this paper we introduce appropriate associated function to the sequence $M_p=p^{\t p^{\s}}$, $p\in \N$, $\t>0$, $\s>1$, and derive its sharp asymptotic estimates in terms of the Lambert $W$ function. These estimates are used to prove a Paley-Wiener type theorem for compactly supported functions from extended Gevrey classes. As an application, we discuss properties of the corresponding wave front sets.

math.FA↗
arXiv2018

Subspace Almost Periodic $C$-Distribution Semigroups and $C$-Distribution Cosine Functions

The main aim of this paper is to introduce and analyze the notions of subspace almost periodicity and subspace weak almost periodicity for $C$-distribution semigroups and $C$-distribution cosine functions in Banach spaces. We continue our previous research study of almost periodicity of abstract Volterra integro-differential equations \cite{aot}, focusing our attention on the abstract ill-posed Cauchy problems of first order.

math.FA↗
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↗
arXiv2017

Frame expansions of test functions, tempered distributions, and ultradistributions

The paper is devoted to frame expansions in Fréchet spaces. First we review some results which concern series expansions in general Fréchet spaces via Fréchet and General Fréchet frames. Then we present some new results on series expansions of tempered distributions and ultradistributions, and the corresponding test functions, via localized frames and coefficients in appropriate sequence spaces.

math.FA↗