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Fedor Sukochev - One of the best experts on this subject based on the ideXlab platform.
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Extreme points of the set of elements majorised by an Integrable Function: Resolution of a problem by Luxemburg and of its noncommutative counterpart
Advances in Mathematics, 2020Co-Authors: D. Dauitbek, Jinghao Huang, Fedor SukochevAbstract:Abstract Let f be an arbitrary Integrable Function on a finite measure space ( X , Σ , ν ) . We characterise the extreme points of the set Ω ( f ) of all measurable Functions on ( X , Σ , ν ) majorised by f, providing a complete answer to a problem raised by W.A.J. Luxemburg in 1967. Moreover, we obtain a noncommutative version of this result.
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Extreme points of the set of elements majorised by an Integrable Function: Resolution of a problem by Luxemburg and of its noncommutative counterpart
arXiv: Functional Analysis, 2019Co-Authors: D. Dauitbek, Jinghao Huang, Fedor SukochevAbstract:Let $f$ be an arbitrary Integrable Function on a finite measure space $(X,\Sigma, \nu)$. We characterise the extreme points of the set $\Omega (f)$ of all measurable Functions on $(X,\Sigma, \nu)$ majorised by $f$, providing a complete answer to a problem raised by W.A.J. Luxemburg in 1967. Moreover, we obtain a noncommutative version of this result.
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Extreme points of the set of elements majorised by an Integrable Function: Resolution of a problem by Luxemburg
arXiv: Functional Analysis, 2019Co-Authors: D. Dauitbek, Jinghao Huang, Fedor SukochevAbstract:Let $f$ be an arbitrary Integrable Function on a finite measure space $(X,\Sigma, \nu)$. We characterise the extreme points of the set $\Omega (f)$ of all measurable Functions on $(X,\Sigma, \nu)$ majorised by $f$, providing a complete answer to a problem raised by W.A.J. Luxemburg in 1967. Moreover, we obtain a noncommutative version of this result.
Valery Serov - One of the best experts on this subject based on the ideXlab platform.
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the fourier series of a square Integrable Function the riesz fischer theorem
2017Co-Authors: Valery SerovAbstract:The set of square-Integrable Functions \(L^2(-\pi ,\pi )\) is an inner product space (linear Euclidean space) equipped with the inner product
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The Fourier Series of a Square-Integrable Function. The Riesz–Fischer Theorem.
Applied Mathematical Sciences, 2017Co-Authors: Valery SerovAbstract:The set of square-Integrable Functions \(L^2(-\pi ,\pi )\) is an inner product space (linear Euclidean space) equipped with the inner product
Dai Feng - One of the best experts on this subject based on the ideXlab platform.
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strong approximation by fourier laplace series in l norm
Advances in Mathematics, 2004Co-Authors: Dai FengAbstract:Let f be an Integrable Function on the unit sphere∑n-1 of Rn (n ≥3) and let SσNδ(f) be the translation operator with step δ ∈R. Let σNδ (f) be the Cesaro means of order S of the Fourier Laplace series of f. This paper proves that if, then and,where Eκλ(f) is the equiconvergent operator of Cesaro means σκλ?
Joao Nuno Prata - One of the best experts on this subject based on the ideXlab platform.
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what is the wigner Function closest to a given square Integrable Function
Siam Journal on Mathematical Analysis, 2018Co-Authors: Jonathan S Benbenjamin, Nuno Costa Dias, Patrick J Loughlin, Leon Cohen, Joao Nuno PrataAbstract:We consider an arbitrary square Integrable Function $F$ on the phase-space and look for the Wigner Function closest to it with respect to the $L^2$ norm. It is well known that the minimizing solution is the Wigner Function of any eigenvector associated with the largest eigenvalue of the Hilbert--Schmidt operator with Weyl symbol $F$. We solve the particular case of radial Functions on the two-dimensional phase space exactly. For more general cases, one has to solve an infinite dimensional eigenvalue problem. To avoid this difficulty, we consider a finite dimensional approximation and estimate the errors for the eigenvalues and eigenvectors. As an application, we address the so-called Wigner approximation suggested by some of us for the propagation of a pulse in a general dispersive medium. We prove that this approximation never leads to a bona fide Wigner Function. This is our prime motivation for our optimization problem. As a by-product of our results, we are able to estimate the eigenvalues and Schatte...
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what is the wigner Function closest to a given square Integrable Function
Siam Journal on Mathematical Analysis, 2018Co-Authors: Jonathan S Benbenjamin, Nuno Costa Dias, Patrick J Loughlin, Leon Cohen, Joao Nuno PrataAbstract:We consider an arbitrary square Integrable Function $F$ on the phase-space and look for the Wigner Function closest to it with respect to the $L^2$ norm. It is well known that the minimizing soluti...
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what is the wigner Function closest to a given square Integrable Function
arXiv: Mathematical Physics, 2018Co-Authors: Jonathan S Benbenjamin, Nuno Costa Dias, Patrick J Loughlin, Leon Cohen, Joao Nuno PrataAbstract:We consider an arbitrary square Integrable Function $F$ on the phase space and look for the Wigner Function closest to it with respect to the $L^2$ norm. It is well known that the minimizing solution is the Wigner Function of any eigenvector associated with the largest eigenvalue of the Hilbert-Schmidt operator with Weyl symbol $F$. We solve the particular case of radial Functions on the two-dimensional phase space exactly. For more general cases, one has to solve an infinite dimensional eigenvalue problem. To avoid this difficulty, we consider a finite dimensional approximation and estimate the errors for the eigenvalues and eigenvectors. As an application, we address the so-called Wigner approximation suggested by some of us for the propagation of a pulse in a general dispersive medium. We prove that this approximation never leads to a {\it bona fide} Wigner Function. This is our prime motivation for our optimization problem. As a by-product of our results we are able to estimate the eigenvalues and Schatten norms of certain Schatten-class operators. The techniques presented here may be potentially interesting for estimating eigenvalues of localization operators in time-frequency analysis and quantum mechanics.
Thai Thuan Quang - One of the best experts on this subject based on the ideXlab platform.
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norm of the hausdorff operator on the real hardy space h 1 mathbb r
Complex Analysis and Operator Theory, 2018Co-Authors: Ha Duy Hung, Thai Thuan QuangAbstract:Let \(\varphi \) be a nonnegative Integrable Function on \((0,\infty )\). It is well-known that the Hausdorff operator \({{\mathcal {H}}}_\varphi \) generated by \(\varphi \) is bounded on the real Hardy space \(H^1({{\mathbb {R}}})\). The aim of this paper is to give the exact norm of \({{\mathcal {H}}}_\varphi \). More precisely, we prove that $$\begin{aligned} \Vert {{\mathcal {H}}}_\varphi \Vert _{H^1({{\mathbb {R}}})\rightarrow H^1({{\mathbb {R}}})}= {\mathop {\int }\limits _{0}^{\infty }} \varphi (t)dt. \end{aligned}$$
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norm of the hausdorff operator on the real hardy space h 1 mathbb r
arXiv e-prints, 2017Co-Authors: Ha Duy Hung, Thai Thuan QuangAbstract:Let $\varphi$ be a nonnegative Integrable Function on $(0,\infty)$. It is well-known that the Hausdorff operator $\mathcal H_\varphi$ generated by $\varphi$ is bounded on the real Hardy space $H^1(\mathbb R)$. The aim of this paper is to give the exact norm of $\mathcal H_\varphi$. More precisely, we prove that $$\|\mathcal H_\varphi\|_{H^1(\mathbb R)\to H^1(\mathbb R)}= \int_0^\infty \varphi(t)dt.$$