Characterizations of g-frames and g-Riesz bases in Hilbert spaces

2008 ◽  
Vol 24 (10) ◽  
pp. 1727-1736 ◽  
Author(s):  
Yu Can Zhu
Keyword(s):  
Hilbert Spaces ◽  
Riesz Bases

scholarly journals GENERAL FRAMEWORK FOR CONSISTENT SAMPLING IN HILBERT SPACES

2005 ◽  
Vol 03 (04) ◽  
pp. 497-509 ◽  
Author(s):  
YONINA C. ELDAR ◽  
TOBIAS WERTHER
Keyword(s):  
Hilbert Spaces ◽  
General Framework ◽  
Inner Product ◽  
Riesz Bases ◽  
Oblique Projection ◽  
Dual Frames ◽  
Infinite Dimensional ◽  
Reconstruction Scheme ◽  
Linear Reconstruction ◽  
Oblique Dual

We introduce a general framework for consistent linear reconstruction in infinite-dimensional Hilbert spaces. We study stable reconstructions in terms of Riesz bases and frames, and generalize the notion of oblique dual frames to infinite-dimensional frames. As we show, the linear reconstruction scheme coincides with the so-called oblique projection, which turns into an ordinary orthogonal projection when adapting the inner product. The inner product of interest is, in general, not unique. We characterize the inner products and corresponding positive operators for which the new geometrical interpretation applies.

scholarly journals GENERAL FRAMEWORK FOR CONSISTENT SAMPLING IN HILBERT SPACES

2005 ◽  
Vol 03 (03) ◽  
pp. 347-359 ◽  
Author(s):  
YONINA C. ELDAR ◽  
TOBIAS WERTHER
Keyword(s):  
Hilbert Spaces ◽  
General Framework ◽  
Inner Product ◽  
Riesz Bases ◽  
Oblique Projection ◽  
Dual Frames ◽  
Infinite Dimensional ◽  
Reconstruction Scheme ◽  
Linear Reconstruction ◽  
Oblique Dual

We introduce a general framework for consistent linear reconstruction in infinite-dimensional Hilbert spaces. We study stable reconstructions in terms of Riesz bases and frames, and generalize the notion of oblique dual frames to infinte-dimensional frames. As we show, the linear reconstruction scheme coincides with the so-called oblique projection, which turns into an ordinary orthogonal projection when adapting the inner product. The inner product of interest is, in general, not unique. We characterize the inner products and the corresponding positive operators for which this geometrical interpretation applies.

scholarly journals Decompositions ofg-Frames and Duals and Pseudoduals ofg-Frames in Hilbert Spaces

2015 ◽  
Vol 2015 ◽  
pp. 1-7
Author(s):  
Xunxiang Guo
Keyword(s):  
Hilbert Spaces ◽  
Riesz Bases ◽  
Bounded Operators ◽  
Linear Combinations ◽  
Orthonormal Bases ◽  
Critical Components

Firstly, we study the representation ofg-frames in terms of linear combinations of simpler ones such asg-orthonormal bases,g-Riesz bases, and normalized tightg-frames. Then, we study the dual and pseudodual ofg-frames, which are critical components in reconstructions. In particular, we characterize the dualg-frames in a constructive way; that is, the formulae for dualg-frames are given. We also give someg-frame like representations for pseudodualg-frame pairs. The operator parameterizations ofg-frames and decompositions of bounded operators are the key tools to prove our main results.

scholarly journals $\boldsymbol{K}$-frames and $\boldsymbol{K}$-Riesz bases incomplex Hilbert spaces

2018 ◽  
Vol 48 (5) ◽  
pp. 609 ◽  
Author(s):  
Zhu Yucan ◽  
Shu Zhibiao ◽  
Xiao Xiangchun
Keyword(s):  
Hilbert Spaces ◽  
Riesz Bases

g-FRAMES AND MODULAR RIESZ BASES IN HILBERT C*-MODULES

2012 ◽  
Vol 10 (02) ◽  
pp. 1250013 ◽  
Author(s):  
AMIR KHOSRAVI ◽  
BEHROOZ KHOSRAVI
Keyword(s):  
Hilbert Spaces ◽  
Riesz Basis ◽  
Riesz Bases ◽  
Parseval Frame ◽  
C Algebra ◽  
Perturbation Result ◽  
Natural Way

In this paper we introduce modular Riesz basis, modular g-Riesz basis in Hilbert C*-modules in a very natural way and we show that they share many properties with Riesz basis and g-Riesz basis in Hilbert spaces. We also found that by using the fact that every finitely or countably generated Hilbert C*-module over a unital C*-algebra has a standard Parseval frame, we characterize g-frames, modular Riesz bases and modular g-Riesz bases. Finally we obtain a perturbation result for modular g-Riesz bases.

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.

C∗-algebra consisting of all g-Bessel sequences in a Hilbert space

2017 ◽  
Vol 15 (01) ◽  
pp. 1750004
Author(s):  
Z. L. Chen ◽  
H. X. Cao ◽  
Z. H. Guo
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Small Perturbation ◽  
Hilbert Spaces ◽  
Operator Theory ◽  
Operator Space ◽  
Riesz Bases ◽  
C Algebra ◽  
Linear Bijection ◽  
Bessel Sequences

For Hilbert spaces [Formula: see text] and [Formula: see text], we use the notations [Formula: see text], [Formula: see text] and [Formula: see text] to denote the sets of all [Formula: see text]-Bessel sequences, [Formula: see text]-frames and Riesz bases in [Formula: see text] with respect to [Formula: see text], respectively. By defining a linear operation and a norm, we prove that [Formula: see text] becomes a Banach space and is isometrically isomorphic to the operator space [Formula: see text], where [Formula: see text]. In light of operator theory, it is proved that [Formula: see text] and [Formula: see text] are open sets in [Formula: see text]. This implies that both [Formula: see text]-frames and Riesz bases are stable under a small perturbation. By introducing a linear bijection [Formula: see text] from [Formula: see text] onto the [Formula: see text]-algebra [Formula: see text], a multiplication and an involution on the Banach space [Formula: see text] are defined so that [Formula: see text] becomes a unital [Formula: see text]-algebra that is isometrically isomorphic to the [Formula: see text]-algebra [Formula: see text], provided that [Formula: see text] and [Formula: see text] are isomorphic.

HILBERT–SCHMIDT OPERATORS AND FRAMES — CLASSIFICATION, BEST APPROXIMATION BY MULTIPLIERS AND ALGORITHMS

2008 ◽  
Vol 06 (02) ◽  
pp. 315-330 ◽  
Author(s):  
PETER BALAZS
Keyword(s):  
Hilbert Spaces ◽  
Best Approximation ◽  
Inner Product ◽  
Riesz Bases ◽  
Wavelet Frames ◽  
Arbitrary Matrix ◽  
The Best Approximation ◽  
Rank One ◽  
Arbitrary Operator ◽  
Bessel Sequences

In this paper we deal with the theory of Hilbert–Schmidt operators, when the usual choice of orthonormal basis, on the associated Hilbert spaces, is replaced by frames. We More precisely, we provide a necessary and sufficient condition for an operator to be Hilbert–Schmidt, based on its action on the elements of a frame (i.e. an operator T is [Formula: see text] if and only if the sum of the squared norms of T applied on the elements of the frame is finite). Also, we construct Bessel sequences, frames and Riesz bases of [Formula: see text] operators using tensor products of the same sequences in the associated Hilbert spaces. We state how the [Formula: see text] inner product of an arbitrary operator and a rank one operator can be calculated in an efficient way; and we use this result to provide a numerically efficient algorithm to find the best approximation, in the Hilbert–Schmidt sense, of an arbitrary matrix, by a so-called frame multiplier (i.e. an operator which act diagonally on the frame analysis coefficients). Finally, we give some simple examples using Gabor and wavelet frames, introducing in this way wavelet multipliers.

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