scholarly journals Hilbert space valued Gabor frames in weighted amalgam spaces

2019 ◽  
Vol 10 (4) ◽  
pp. 377-394
Author(s):  
Anirudha Poria ◽  
Jitendriya Swain
Keyword(s):  
Hilbert Space ◽  
Separable Hilbert Space ◽  
Gabor Frame ◽  
Window Function ◽  
Gabor Frames ◽  
Frame Operator ◽  
Wiener Amalgam Space ◽  
Amalgam Spaces ◽  
Special Case

AbstractLet {\mathbb{H}} be a separable Hilbert space. In this paper, we establish a generalization of Walnut’s representation and Janssen’s representation of the {\mathbb{H}}-valued Gabor frame operator on {\mathbb{H}}-valued weighted amalgam spaces {W_{\mathbb{H}}(L^{p},L^{q}_{v})}, {1\leq p,q\leq\infty}. Also, we show that the frame operator is invertible on {W_{\mathbb{H}}(L^{p},L^{q}_{v})}, {1\leq p,q\leq\infty}, if the window function is in the Wiener amalgam space {W_{\mathbb{H}}(L^{\infty},L^{1}_{w})}. Further, we obtain the Walnut representation and invertibility of the frame operator corresponding to Gabor superframes and multi-window Gabor frames on {W_{\mathbb{H}}(L^{p},L^{q}_{v})}, {1\leq p,q\leq\infty}, as a special case by choosing the appropriate Hilbert space {\mathbb{H}}.

scholarly journals Invertibility of the Gabor frame operator on the Wiener amalgam space

2008 ◽  
Vol 153 (2) ◽  
pp. 212-224 ◽  
Author(s):  
Ilya A. Krishtal ◽  
Kasso A. Okoudjou
Keyword(s):  
Gabor Frame ◽  
Frame Operator ◽  
Wiener Amalgam Space

scholarly journals On the Parity Under Metapletic Operators and an Extension of a Result of Lyubarskii and Nes

2019 ◽  
Vol 75 (1) ◽  
Author(s):  
Markus Faulhuber
Keyword(s):  
Hilbert Space ◽  
Real Line ◽  
Gabor Frame ◽  
Window Function ◽  
Gabor System ◽  
The Real ◽  
Metaplectic Group ◽  
Integrable Functions ◽  
Square Integrable Functions

AbstractIn this work we show that if the frame property of a Gabor frame with window in Feichtinger’s algebra and a fixed lattice only depends on the parity of the window, then the lattice can be replaced by any other lattice of the same density without losing the frame property. As a byproduct we derive a generalization of a result of Lyubarskii and Nes, who could show that any Gabor system consisting of an odd window function from Feichtinger’s algebra and any separable lattice of density $$\tfrac{n+1}{n}$$n+1n, $$n \in \mathbb {N}_+$$n∈N+, cannot be a Gabor frame for the Hilbert space of square-integrable functions on the real line. We extend this result by removing the assumption that the lattice has to be separable. This is achieved by exploiting the interplay between the symplectic and the metaplectic group.

A Cohomological Property of π-invariant Elements

2013 ◽  
Vol 56 (3) ◽  
pp. 534-543
Author(s):  
M. Filali ◽  
M. Sangani Monfared
Keyword(s):  
Hilbert Space ◽  
Banach Algebra ◽  
Orthonormal Basis ◽  
Hochschild Cohomology ◽  
Separable Hilbert Space ◽  
Continuous Representation ◽  
Cohomology Groups ◽  
Cohomological Property ◽  
Coordinate Functions ◽  
Special Case

Abstract.Let A be a Banach algebra and let be a continuous representation of A on a separable Hilbert space H with dim H = m. Let πi j be the coordinate functions of π with respect to an orthonormal basis and suppose that for each and . Under these conditions, we call an element left π-invariant if In this paper we prove a link between the existence of left π-invariant elements and the vanishing of certain Hochschild cohomology groups of A. Our results extend an earlier result by Lau on F-algebras and recent results of Kaniuth, Lau, Pym, and and the second author in the special case where π : A → C is a non-zero character on A.

scholarly journals WEIGHTED AND CONTROLLED FRAMES: MUTUAL RELATIONSHIP AND FIRST NUMERICAL PROPERTIES

2010 ◽  
Vol 08 (01) ◽  
pp. 109-132 ◽  
Author(s):  
PETER BALAZS ◽  
JEAN-PIERRE ANTOINE ◽  
ANNA GRYBOŚ
Keyword(s):  
Iterative Algorithms ◽  
Gabor Frames ◽  
Dual Frame ◽  
Mutual Relationship ◽  
Frame Operator ◽  
Numerical Efficiency ◽  
Frame Condition ◽  
Frame Bound ◽  
Special Case ◽  
Numerical Advantage

Weighted and controlled frames have been introduced recently to improve the numerical efficiency of iterative algorithms for inverting the frame operator. In this paper, we develop systematically these notions, including their mutual relationship. We will show that controlled frames are equivalent to standard frames and so this concept gives a generalized way to check the frame condition, while offering a numerical advantage in the sense of preconditioning. Next, we investigate weighted frames, in particular their relation to controlled frames. We consider the special case of semi-normalized weights, where the concepts of weighted frames and standard frames are interchangeable. We also make the connection with frame multipliers. Finally, we analyze weighted frames numerically. First, we investigate three possibilities for finding weights in order to tighten a given frame, i.e. decrease the frame bound ratio. Then, we examine Gabor frames and how well the canonical dual of a weighted frame is approximated by the inversely weighted dual frame.

scholarly journals Behavior of Gabor frame operators on Wiener amalgam spaces

2016 ◽  
Vol 14 (04) ◽  
pp. 1650028 ◽  
Author(s):  
Anirudha Poria
Keyword(s):  
Weak Sense ◽  
Gabor Frame ◽  
Operator Norm ◽  
Identity Operator ◽  
Frame Operator ◽  
Wiener Amalgam Spaces ◽  
Sampling Density ◽  
Amalgam Spaces

It is well-known that the Gabor expansions converge to identity operator in weak* sense on the Wiener amalgam spaces as sampling density tends to infinity. In this paper, we prove the convergence of Gabor expansions to identity operator in the operator norm as well as weak* sense on [Formula: see text] as the sampling density tends to infinity. Also we show the validity of the Janssen’s representation and the Wexler–Raz biorthogonality condition for Gabor frame operator on [Formula: see text].

scholarly journals A note on local asymptotic behaviour for Brownian motion in Banach spaces

1979 ◽  
Vol 2 (4) ◽  
pp. 669-676
Author(s):  
Mou-Hsiung Chang
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Brownian Motion ◽  
Gaussian Measure ◽  
Separable Hilbert Space ◽  
Integral Test ◽  
Real Separable Banach Space ◽  
Real Separable Hilbert Space ◽  
Special Case

In this paper we obtain an integral characterization of a two-sided upper function for Brownian motion in a real separable Banach space. This characterization generalizes that of Jain and Taylor [2] whereB=ℝn. The integral test obtained involves the index of a mean zero Gaussian measure on the Banach space, which is due to Kuelbs [3]. The special case that whenBis itself a real separable Hilbert space is also illustrated.

scholarly journals Absorption in Invariant Domains for Semigroups of Quantum Channels

2021 ◽  
Author(s):  
Raffaella Carbone ◽  
Federico Girotti
Keyword(s):  
Hilbert Space ◽  
Ergodic Theory ◽  
Markov Processes ◽  
Separable Hilbert Space ◽  
Quantum Channels ◽  
Quantum Markov Processes ◽  
Positive Recurrent ◽  
Markov Evolution ◽  
Absorption Probabilities ◽  
Invariant Domains

AbstractWe introduce a notion of absorption operators in the context of quantum Markov processes. The absorption problem in invariant domains (enclosures) is treated for a quantum Markov evolution on a separable Hilbert space, both in discrete and continuous times: We define a well-behaving set of positive operators which can correspond to classical absorption probabilities, and we study their basic properties, in general, and with respect to accessibility structure of channels, transience and recurrence. In particular, we can prove that no accessibility is allowed between the null and positive recurrent subspaces. In the case, when the positive recurrent subspace is attractive, ergodic theory will allow us to get additional results, in particular about the description of fixed points.

Functional Quantum Theory of Free Relativistic Fermi Fields

1970 ◽  
Vol 25 (5) ◽  
pp. 575-586
Author(s):  
H. Stumpf
Keyword(s):  
Hilbert Space ◽  
Quantum Theory ◽  
Physical Interpretation ◽  
Functional Equations ◽  
Dirac Field ◽  
Functional Hilbert Space ◽  
Free Fermi ◽  
Special Case ◽  
Relativistic Fermi

Functional quantum theory of free Fermi fields is treated for the special case of a free Dirac field. All other cases run on the same pattern. Starting with the Schwinger functionals of the free Dirac field, functional equations and corresponding many particle functionals can be derived. To establish a functional quantum theory, a physical interpretation of the functionals is required. It is provided by a mapping of the physical Hilbert space into an appropriate functional Hilbert space, which is introduced here. Mathematical details, especially the problems connected with anticommuting functional sources are treated in the appendices.

Sign in / Sign up

Export Citation Format

Share Document