The Experts below are selected from a list of 312 Experts worldwide ranked by ideXlab platform
Joydip Jana - One of the best experts on this subject based on the ideXlab platform.
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Image of the Schwartz Space under spectral projection
Mathematische Zeitschrift, 2012Co-Authors: Joydip JanaAbstract:Let X = G / K be a Riemannian symmetric Space of non compact type and rank-one. The spectral projection P _λ f of a function f on X can be written P _λ f = f * φ _λ where φ _λ is the elementary spherical function corresponding to the complex parameter λ. We characterize the image of the Schwartz Space $${\mathcal S^p(X)}$$ under the spectral projection for 0 < p ≤ 2.
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on the Schwartz Space isomorphism theorem for the riemannian symmetric Spaces
arXiv: Representation Theory, 2010Co-Authors: Joydip JanaAbstract:We deduce a proof of the isomorphism theorem for certain closed subSpace $\mc S^p_\Gamma(X)$ of the $L^p$-Schwartz class functions $(0< p \leq 2) $ on a Riemannian symmetric Space $X$ where $\Gamma$ is a finite subset of $\what{K}_M$. The Fourier transform considered is the Helgason Fourier transform. Our proof relies only on the Paley-Wiener theorem for the corresponding class of functions and hence it does not use the complicated higher asymptotics of the elementary spherical functions.
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Image of Schwartz Space Under Spectral Projection
arXiv: Representation Theory, 2007Co-Authors: Joydip JanaAbstract:Let $X= G/K$ symmetric Space of non compact type, where $G$ is a rank-one connected semisimple Lie group with finite center. We shall look at the transform $ P_\lambda f(x) = f \ast \varphi_\lambda(x)$, where, $\lambda \in \mathbb C$ and $\varphi_\lambda$ is the elementary spherical function. We shall try to characterizes the image of the Schwartz Spaces $S^p(X) $ where $0 < p \leq 2$ under the above transform.
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on the Schwartz Space isomorphism theorem for rank one symmetric Space
arXiv: Representation Theory, 2007Co-Authors: Joydip Jana, Rudra P SarkarAbstract:In this paper we give a simpler proof of the $L^p$-Schwartz Space isomorphism $(0< p\leq 2)$ under the Fourier transform for the class of functions of left $\delta$-type on a Riemannian symmetric Space of rank one. Our treatment rests on Anker's \cite{A} proof of the corresponding result in the case of left $K$-invariant functions on $X$. Thus we give a proof which relies only on the Paley--Wiener theorem.
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on the Schwartz Space isomorphism theorem for rank one symmetric Space
Proceedings Mathematical Sciences, 2007Co-Authors: Joydip Jana, Rudra P SarkarAbstract:In this paper we give a simpler proof of the Lp-Schwartz Space isomorphism (0 < p ≤ 2) under the Fourier transform for the class of functions of left δ-type on a Riemannian symmetric Space of rank one. Our treatment rests on Anker’s [2] proof of the corresponding result in the case of left K-invariant functions on X. Thus we give a proof which relies only on the Paley-Wiener theorem.
N. F. Abuzyarova - One of the best experts on this subject based on the ideXlab platform.
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Principal Submodules in the Module of Entire Functions, Which is Dual to the Schwarz Space, and Weak Spectral Synthesis in the Schwartz Space
Journal of Mathematical Sciences, 2019Co-Authors: N. F. AbuzyarovaAbstract:We obtain a sufficient condition of the weak localizability of a principal submodule in the module of entire functions of exponential type and polynomial growth on the real line. Applications to the problem of the (weak) spectral synthesis in the Schwartz Space C∞ (a; b) are discussed.
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Spectral synthesis for the differentiation operator in the Schwartz Space
Mathematical Notes, 2017Co-Authors: N. F. AbuzyarovaAbstract:We consider the spectral synthesis problem for the differentiation operator on the Space of infinitely differentiable functions on a finite or infinite interval of the real line and the dual problem of local description of closed submodules in a special module of entire functions.
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Spectral synthesis for the differentiation operator in the Schwartz Space
arXiv: Complex Variables, 2016Co-Authors: N. F. AbuzyarovaAbstract:We consider the spectral synthesis problem for the differentiation operator D=d/dt in the Schwartz Space E(a;b) and the dual problem of local description for closed submodules in a special module of entire functions.
Yiannis Sakellaridis - One of the best experts on this subject based on the ideXlab platform.
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Correction to: The Schwartz Space of a smooth semi-algebraic stack
Selecta Mathematica, 2018Co-Authors: Yiannis SakellaridisAbstract:The purpose of this note is to fix two gaps in the construction of Schwartz Spaces of semi-algebraic stacks in [4], and to strengthen some statements, replacing quasi-isomorphisms by homotopy equivalences.
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The Schwartz Space of a smooth semi-algebraic stack
Selecta Mathematica, 2016Co-Authors: Yiannis SakellaridisAbstract:Schwartz functions, or measures, are defined on any smooth semi-algebraic (“Nash”) manifold, and are known to form a cosheaf for the semi-algebraic restricted topology. We extend this definition to smooth semi-algebraic stacks, which are defined as geometric stacks in the category of Nash manifolds. Moreover, when those are obtained from algebraic quotient stacks of the form X / G , with X a smooth affine variety and G a reductive group defined over a number field k , we define, whenever possible, an “evaluation map” at each semisimple k -point of the stack, without using truncation methods. This corresponds to a regularization of the sum of those orbital integrals whose semisimple part corresponds to the chosen k -point. These evaluation maps produce, in principle, a distribution which generalizes the Arthur–Selberg trace formula and Jacquet’s relative trace formula, although the former, and many instances of the latter, cannot actually be defined by the purely geometric methods of this paper. In any case, the stack-theoretic point of view provides an explanation for the pure inner forms that appear in many versions of the Langlands, and relative Langlands, conjectures.
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the Schwartz Space of a smooth semi algebraic stack
arXiv: Algebraic Geometry, 2015Co-Authors: Yiannis SakellaridisAbstract:Schwartz functions, or measures, are defined on any smooth semi-algebraic ("Nash") manifold, and are known to form a cosheaf for the semi-algebraic restricted topology. We extend this definition to smooth semi-algebraic stacks, which are defined as geometric stacks in the category of Nash manifolds. Moreover, when those are obtained from algebraic quotient stacks of the form X/G, with X a smooth affine variety and G a reductive group defined over a global field k, we define, whenever possible, an "evaluation map" at each semisimple k-point of the stack, without using truncation methods. This corresponds to a regularization of the sum of those orbital integrals whose semisimple part corresponds to the chosen k-point. These evaluation maps produce, in principle, a distribution which generalizes the Arthur-Selberg trace formula and Jacquet's relative trace formula, although the former, and many instances of the latter, cannot actually be defined by the purely geometric methods of this paper. In any case, the stack-theoretic point of view provides an explanation for the pure inner forms that appear in many versions of the Langlands, and relative Langlands, conjectures.
Georg Zimmermann - One of the best experts on this subject based on the ideXlab platform.
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hardy s theorem and the short time fourier transform of Schwartz functions
Journal of The London Mathematical Society-second Series, 2001Co-Authors: Karlheinz Gröchenig, Georg ZimmermannAbstract:The Schwartz Space of rapidly decaying test functions is characterized by the decay of the short-time Fourier transform or cross-Wigner distribution. Then a version of Hardy's theorem is proved for the short-time Fourier transform and for the Wigner distribution.
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Hardy's Theorem and the Short‐Time Fourier Transform of Schwartz Functions
Journal of the London Mathematical Society, 2001Co-Authors: Karlheinz Gröchenig, Georg ZimmermannAbstract:The Schwartz Space of rapidly decaying test functions is characterized by the decay of the short-time Fourier transform or cross-Wigner distribution. Then a version of Hardy's theorem is proved for the short-time Fourier transform and for the Wigner distribution.
Taira Kouichi - One of the best experts on this subject based on the ideXlab platform.
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Scattering theory for repulsive Schr\"odinger operators and applications to limit circle problem
2019Co-Authors: Taira KouichiAbstract:In this note, we study existence of the outgoing/incoming resolvents of repulsive Schr\"odinger operators which may not be essentially self-adjoint on the Schwartz Space. Moreover, we recover the classical result: The repulsive Schro\"odinger operators with large repulsive constant is not essentially self-adjoint on the Schwartz Space.Comment: 20page
