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Nenad Teofanov

Publications and source records attributed to Nenad Teofanov.

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

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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.

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arXiv2020

Subexponential decay and regularity estimates for eigenfunctions of localization operators

We consider time-frequency localization operators $A_a^{φ_1,φ_2}$ with symbols $a$ in the wide weighted modulation space $ M^\infty_{w}(\mathbb{R}^{2d})$, and windows $ φ_1, φ_2 $ in the Gelfand-Shilov space $\mathcal{S}^{\left(1\right)}(\mathbb{R}^{d})$. If the weights under consideration are of ultra-rapid growth, we prove that the eigenfunctions of $A_a^{φ_1,φ_2}$ have appropriate subexponential decay in phase space, i.e. that they belong to the Gefand-Shilov space $ \mathcal{S}^{(γ)} (\mathbb{R}^{d}) $, where the parameter $γ\geq 1 $ is related to the growth of the considered weight. An important role is played by $τ$-pseudodifferential operators $\mathrm{Op}_τ(σ)$. In that direction we show convenient continuity properties of $\mathrm{Op}_τ(σ)$ when acting on weighted modulation spaces. Furthermore, we prove subexponential decay and regularity properties of the eigenfunctions of $\mathrm{Op}_τ(σ)$ when the symbol $σ$ belongs to a modulation space with appropriately chosen weight functions. As a tool we also prove new convolution relations for (quasi-)Banach weighted modulation spaces.

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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.

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arXiv2019

Bilinear pseudo-differential operators with Gevrey-Hörmander symbols

We consider bilinear pseudo-differential operators whose symbols posses Gevrey type regularity and may have a sub-exponential growth at infinity, together with all their derivatives. It is proved that those symbol classes can be described by the means of the short-time Fourier transform and modulation spaces. Our first main result is the invariance property of the corresponding bilinear operators. Furthermore we prove the continuity of such operators when acting on modulation spaces. As a consequence, we derive their continuity on anisotropic Gelfand-Shilov type spaces. We consider both Beurling and Roumieu type symbol classes and Gelfand-Shilov spaces.

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arXiv2019

The pseudo-differential calculus in a Bargmann setting

We give a fundament for Berezin's analytic $Ψ$do considered in \cite{Berezin71} in terms of Bargmann images of Pilipovi{ć} spaces. We deduce basic continuity results for such $Ψ$do, especially when the operator kernels are in suitable mixed weighted Lebesgue spaces and act on certain weighted Lebesgue spaces of entire functions. In particular, we show how these results imply well-known continuity results for real $Ψ$do with symbols in modulation spaces, when acting on other modulation spaces.

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arXiv2019

Extended Gevrey regularity via the short-time Fourier transform

We study the regularity of smooth functions whose derivatives are dominated by sequences of the form $M_p^{τ,\s}=p^{τp^{\s}}$, $τ>0$, $\s\geq1$. We show that such functions can be characterized through the decay properties of their short-time Fourier transforms (STFT), and recover \cite[Theorem 3.1]{CNR} as the special case when $ \t>1$ and $\s = 1$, i.e. when the Gevrey type regularity is considered. These estimates lead to a Paley-Wiener type theorem for extended Gevrey classes. In contrast to the related result from \cite{PTT-05, PTT-04}, here we relax the assumption on compact support of the observed functions. Moreover, we introduce the corresponding wave front set, recover it in terms of the STFT, and discuss local regularity in such context.

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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.

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arXiv2018

Wave-front sets in non-quasianalytic setting for Fourier Lebesgue and modulation spaces

We define and study wave-front sets for weighted Fourier-Lebesgue spaces when the weights are moderate with respect to the associated functions for general sequences $\{ M_p\} $ which satisfy Komatsu's conditions $(M.1) - (M.3)'$. In particular, when $\{ M_p\} $ is the Gevrey sequence ($M_p = p!^s$, $s>1$) we recover some previously observed results. Furthermore, we consider wave-front sets for modulation spaces in the same setting, and prove the invariance property related to the Fourier-Lebesgue type wave-front sets.

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arXiv2018

The Grossmann-Royer transform, Gelfand-Shilov spaces, and continuity properties of localization operators on modulation spaces

This paper offers a review of the results concerning localization operators on modulation spaces, and related topics. However, our approach, based on the Grossmann-Royer transform, gives a new insight and (slightly) different proofs. We define the Grossmann-Royer transform as interpretation of the Grossmann-Royer operator in the weak sense. Although such transform is essentially the same as the cross-Wigner distribution, the proofs of several known results are simplified when it is used instead of other time-frequency representations. Due to the importance of their role in applications when dealing with ultrafast decay properties in phase space, we give a detailed account on the Gelfand-Shilov spaces and their dual spaces, and extend the Grossmann-Royer transform and its properties in such context. Another family of spaces, modulation spaces, are recognized as appropriate background for time-frequency analysis. In particular, the Gelfand-Shilov spaces are revealed as projective and inductive limits of modulation spaces. For the continuity and compactness properties of localization operators we employ the norms in modulation spaces. We define localization operators in terms of the Grossmann-Royer transform, and show that such definition coincides with the usual definition based on the short-time Fourier transform.

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arXiv2018

Continuity properties of multilinear localization operators on modulation spaces

We introduce multilinear localization operators in terms of the short-time Fourier transform, and multilinear Weyl pseudodifferential operators. We prove that such localization operators are in fact Weyl pseudodifferential operators whose symbols are given by the convolution between the symbol of the localization operator and the multilinear Wigner transform. For such interpretation we use the kenrel theorem for the Gelfand-Shilov space. Furthermore, we study the continuity properties of the multilinear localization operators on modulation spaces. Our results extend some known results when restricted to the linear case.

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arXiv2017

Superposition and propagation of singularities for extended Gevrey regularity

We use sequences which depend on two parameters to define families of ultradifferentiable functions which contain Gevrey classes. It is shown that such families are closed under superposition, and therefore inverse closed as well. Furthermore, we study partial differential operators whose coefficients satisfy the extended Gevrey regularity. To that aim we introduce appropriate wave front sets and derive a theorem on propagation of singularities. This extends related known results in the sense that weaker assumptions on the regularity of the coefficients are imposed.

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arXiv2016

Ultradifferentiable functions of class $M_p^{\t,\s}$ and microlocal regularity

We study spaces of ultradifferentiable functions which contain Gevrey classes. Although the corresponding defining sequences do not satisfy Komatsu's condition (M.2)', we prove appropriate continuity properties under the action of (ultra)differentiable operators. Furthermore, we study convenient localization procedure which leads to the concept of wave-front set with respect to our regularity conditions. As an application, we identify the standard projections of intersections/unions of wave-front sets as singular supports of suitable spaces of ultradifferentiable functions.

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arXiv2016

Beyond Gevrey regularity

We define and study classes of smooth functions which are less regular than Gevrey functions. To that end we introduce two-parameter dependent sequences which do not satisfy Komatsu's condition (M.2)', which implies stability under differential operators within the spaces of ultradifferentiable functions. Our classes therefore have particular behavior under the action of differentiable operators. On a more advanced level, we study microlocal properties and prove that $${\rm WF}_{0,\infty}(P(D)u)\subseteq {\rm WF}_{0,\infty}(u)\subseteq {\rm WF}_{0,\infty}(P(D)u) \cup {\rm Char}(P),$$ where $u$ is a Schwartz distribution, $P(D)$ is a partial differential operator with constant coefficients and ${\rm WF}_{0,\infty}$ is the wave front set described in terms of new regularity conditions. For the analysis we introduce particular admissibility condition for sequences of cut-off functions, and a new technical tool called enumeration.

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arXiv2015

The wavelet transforms in Gelfand-Shilov spaces

We describe local and global properties of wavelet transforms of ultradifferentiable functions. The results are given in the form of continuity properties of the wavelet transform on Gelfand-Shilov type spaces and their duals. In particular, we introduce a new family of highly time-scale localized spaces on the upper half-space. We study the wavelet synthesis operator (the left-inverse of the wavelet transform) and obtain the resolution of identity (Calderón reproducing formula) in the context of ultradistributions.

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arXiv2011

Micro-local analysis in some spaces of ultradistributions

In this paper we extend some results from our earlier papers on wave-front sets, concerning wave-front sets of Fourier-Lebesgue and modulation space types, to a broader class of spaces of ultradistributions, and relate these wave-front sets with the usual wave-front sets of ultradistributions. Furthermore, we use Gabor frames for the description of discrete wave-front sets, and prove that these wave-front sets coincide with corresponding continuous ones.

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