PERTURBED FRAME SEQUENCES: CANONICAL DUAL SYSTEMS, APPROXIMATE RECONSTRUCTIONS AND APPLICATIONS

2014 ◽  
Vol 12 (02) ◽  
pp. 1450019 ◽  
Author(s):  
SIGRID HEINEKEN ◽  
EWA MATUSIAK ◽  
VICTORIA PATERNOSTRO
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Estimation Error ◽  
Dual Systems ◽  
Approximation Of Functions ◽  
Small Perturbations ◽  
Bandlimited Function ◽  
Frame Sequence

We consider perturbation of frames and frame sequences in a Hilbert space ℋ. It is known that small perturbations of a frame give rise to another frame. We show that the canonical dual of the perturbed sequence is a perturbation of the canonical dual of the original one and estimate the error in the approximation of functions belonging to the perturbed space. We then construct perturbations of irregular translates of a bandlimited function in L2(ℝd). We give conditions for the perturbed sequence to inherit the property of being Riesz or frame sequence. For this case we again calculate the error in the approximation of functions that belong to the perturbed space and compare it with our previous estimation error for general Hilbert spaces.

Generalized frames for operators in Hilbert spaces

2014 ◽  
Vol 17 (02) ◽  
pp. 1450013 ◽  
Author(s):  
Mohammad Sadegh Asgari ◽  
Hamidreza Rahimi
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Separable Hilbert Space ◽  
Closed Subspace ◽  
Bounded Operator ◽  
Riesz Bases ◽  
Small Perturbations ◽  
Riesz Decomposition ◽  
Analysis And Synthesis ◽  
General Range

In this paper we present a family of analysis and synthesis systems of operators with frame-like properties for the range of a bounded operator on a separable Hilbert space. This family of operators is called a Θ–g-frame, where Θ is a bounded operator on a Hilbert space. Θ–g-frames are a generalization of g-frames, which allows to reconstruct elements from the range of Θ. In general, range of Θ is not a closed subspace. We also construct new Θ–g-frames by considering Θ–g-frames for its components. We further study Riesz decompositions for Hilbert spaces, which are a generalization of the notion of Riesz bases. We define the coefficient operators of a Riesz decomposition and we will show that these coefficient operators are continuous projections. We obtain some results about stability of Riesz decompositions under small perturbations.

scholarly journals Translation and modulation invariant Hilbert spaces

2021 ◽  
Author(s):  
Joachim Toft ◽  
Anupam Gumber ◽  
Ramesh Manna ◽  
P. K. Ratnakumar
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Zero Element ◽  
Multiplicative Constant ◽  
Space Of Distributions ◽  
Feichtinger Algebra

AbstractLet $$\mathcal H$$ H be a Hilbert space of distributions on $$\mathbf{R}^{d}$$ R d which contains at least one non-zero element of the Feichtinger algebra $$S_0$$ S 0 and is continuously embedded in $$\mathscr {D}'$$ D ′ . If $$\mathcal H$$ H is translation and modulation invariant, also in the sense of its norm, then we prove that $$\mathcal H= L^2$$ H = L 2 , with the same norm apart from a multiplicative constant.

scholarly journals Clock-dependent spacetime

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Sung-Sik Lee
Keyword(s):  
Hilbert Space ◽  
General Relativity ◽  
Quantum Gravity ◽  
Hilbert Spaces ◽  
Coordinate Systems ◽  
Physical Laws

Abstract Einstein’s theory of general relativity is based on the premise that the physical laws take the same form in all coordinate systems. However, it still presumes a preferred decomposition of the total kinematic Hilbert space into local kinematic Hilbert spaces. In this paper, we consider a theory of quantum gravity that does not come with a preferred partitioning of the kinematic Hilbert space. It is pointed out that, in such a theory, dimension, signature, topology and geometry of spacetime depend on how a collection of local clocks is chosen within the kinematic Hilbert space.

scholarly journals A characterisation of Hilbert spaces via orthogonality and proximinality

2005 ◽  
Vol 71 (1) ◽  
pp. 107-111
Author(s):  
Fathi B. Saidi
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Real Banach Space ◽  
Nonzero Element ◽  
Dimensional Subspace ◽  
Two Dimensional ◽  
Orthogonal Direction ◽  
Open Problems

In this paper we adopt the notion of orthogonality in Banach spaces introduced by the author in [6]. There, the author showed that in any two-dimensional subspace F of E, every nonzero element admits at most one orthogonal direction. The problem of existence of such orthogonal direction was not addressed before. Our main purpose in this paper is the investigation of this problem in the case where E is a real Banach space. As a result we obtain a characterisation of Hilbert spaces stating that, if in every two-dimensional subspace F of E every nonzero element admits an orthogonal direction, then E is isometric to a Hilbert space. We conclude by presenting some open problems.

scholarly journals Some Hilbert spaces related with the Dirichlet space

2016 ◽  
Vol 3 (1) ◽  
Author(s):  
Nicola Arcozzi ◽  
Pavel Mozolyako ◽  
Karl-Mikael Perfekt ◽  
Stefan Richter ◽  
Giulia Sarfatti
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Dirichlet Space

AbstractWe study the reproducing kernel Hilbert space with kernel k

scholarly journals Isometric Group Actions on Hilbert Spaces: Structure of Orbits

2008 ◽  
Vol 60 (5) ◽  
pp. 1001-1009 ◽  
Author(s):  
Yves de Cornulier ◽  
Romain Tessera ◽  
Alain Valette
Keyword(s):  
Hilbert Space ◽  
Nilpotent Group ◽  
Hilbert Spaces ◽  
Metabelian Group ◽  
Isometric Action ◽  
Finitely Generated ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Isometric Group Actions ◽  
Dense Orbits

AbstractOur main result is that a finitely generated nilpotent group has no isometric action on an infinite-dimensional Hilbert space with dense orbits. In contrast, we construct such an action with a finitely generated metabelian group.

scholarly journals On invex programming problem in Hilbert spaces

2015 ◽  
Vol 25 (3) ◽  
pp. 379-385
Author(s):  
Sandip Chatterjee ◽  
Rathindranath Mukherjee
Keyword(s):  
Hilbert Space ◽  
Programming Problem ◽  
Hilbert Spaces

In this paper we introduce the invex programming problem in Hilbert space. The requisite theory has been established to characterize the solution of such class of problems.

scholarly journals Parseval's equality in fuzzy normed linear spaces

Mathematica ◽  
2021 ◽  
Vol 63 (86) (1) ◽  
pp. 47-57
Author(s):  
Daraby Bayaz ◽  
Delzendeh Fataneh ◽  
Rahimi Asghar
Keyword(s):  
Hilbert Space ◽  
Vector Space ◽  
Hilbert Spaces ◽  
Continuous Functions ◽  
Linear Spaces ◽  
Normed Linear Spaces ◽  
Parseval’S Equality

We investigate Parseval's equality and define the fuzzy frame on Felbin fuzzy Hilbert spaces. We prove that C(Omega) (the vector space of all continuous functions on Omega) is normable in a Felbin fuzzy Hilbert space and so defining fuzzy frame on C(Omega) is possible. The consequences for the category of fuzzy frames in Felbin fuzzy Hilbert spaces are wider than for the category of the frames in the classical Hilbert spaces.

A THEOREM OF SOLÉR, THE THEORY OF SYMMETRY AND QUANTUM MECHANICS

2012 ◽  
Vol 09 (02) ◽  
pp. 1260005 ◽  
Author(s):  
GIANNI CASSINELLI ◽  
PEKKA LAHTI
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum Logic ◽  
Hilbert Spaces ◽  
Classical Problem ◽  
Partial Solution ◽  
Space Realization ◽  
Symmetry Transformations ◽  
Axiomatic Quantum Mechanics

A classical problem in axiomatic quantum mechanics is deducing a Hilbert space realization for a quantum logic that admits a vector space coordinatization of the Piron–McLaren type. Our aim is to show how a theorem of M. Solér [Characterization of Hilbert spaces by orthomodular spaces, Comm. Algebra23 (1995) 219–243.] can be used to get a (partial) solution of this problem. We first derive a generalization of the Wigner theorem on symmetry transformations that holds already in the Piron–McLaren frame. Then we investigate which conditions on the quantum logic allow the use of Solér's theorem in order to obtain a Hilbert space solution for the coordinatization problem.

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