scholarly journals Controlled $g$-dual frames and their approximate in Hilbert spaces

2021 ◽  
pp. 1-9
Author(s):  
Sayyed Mehrab RAMEZANİ
Keyword(s):  
Hilbert Spaces ◽  
Dual Frames

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.

Duals and multipliers of controlled frames in Hilbert spaces

2018 ◽  
Vol 16 (06) ◽  
pp. 1850057 ◽  
Author(s):  
M. Rashidi-Kouchi ◽  
A. Rahimi ◽  
Firdous A. Shah
Keyword(s):  
Hilbert Spaces ◽  
Dual Frames ◽  
Multiplier Operator ◽  
Mild Conditions ◽  
Multiplier Operators ◽  
Approximate Duality

In this paper, we introduce and characterize controlled dual frames in Hilbert spaces. We also investigate the relation between bounds of controlled frames and their related frames. Then, we define the concept of approximate duality for controlled frames in Hilbert spaces. Next, we introduce multiplier operators of controlled frames in Hilbert spaces and investigate some of their properties. Finally, we show that the inverse of a controlled multiplier operator is also a controlled multiplier operator under some mild conditions.

scholarly journals Frame for operators in finite dimensional hilbert space

2018 ◽  
Vol 49 (1) ◽  
pp. 35-48
Author(s):  
Mohammad Janfada ◽  
Vahid Reza Morshedi ◽  
Rajabali Kamyabi Gol
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Dual Frames ◽  
Finite Dimensional ◽  
Dimensional Hilbert Space ◽  
Finite Dimensional Hilbert Space

In this paper, we study frames for operators ($K$-frames) in finite dimensional Hilbert spaces and express the dual of $K$-frames. Some properties of $K$-dual frames are investigated. Furthermore, the notion of their oblique $K$-dual and some properties are presented.

DUAL PAIRS OF GABOR FRAMES FOR TRIGONOMETRIC GENERATORS WITHOUT THE PARTITION OF UNITY PROPERTY

2011 ◽  
Vol 04 (04) ◽  
pp. 589-603 ◽  
Author(s):  
O. Christensen ◽  
Mads Sielemann Jakobsen
Keyword(s):  
Hilbert Spaces ◽  
Partition Of Unity ◽  
Frame Theory ◽  
Gabor Frames ◽  
Series Expansions ◽  
Dual Frames ◽  
Dual Pairs ◽  
Orthonormal Bases ◽  
Explicit Constructions

Frames is a strong tool to obtain series expansions in Hilbert spaces under less restrictive conditions than imposed by orthonormal bases. In order to apply frame theory it is necessary to construct a pair of so called dual frames. The goal of the article is to provide explicit constructions of dual pairs of frames having Gabor structure. Unlike the results presented in the literature we do not base the constructions on a generator satisfying the partition of unity constraint.

OBLIQUE DUAL FRAMES IN FINITE-DIMENSIONAL HILBERT SPACES

2013 ◽  
Vol 11 (02) ◽  
pp. 1350011 ◽  
Author(s):  
XIANG-CHUN XIAO ◽  
YU-CAN ZHU ◽  
XIAO-MING ZENG
Keyword(s):  
Hilbert Spaces ◽  
Computational Efficiency ◽  
Tight Frame ◽  
Constructive Method ◽  
Dual Frame ◽  
Dual Frames ◽  
Finite Dimensional ◽  
Oblique Dual

A frame can be completed to a tight frame by adding some additional vectors. However, for the purpose of computational efficiency, we need to put restrictions to the number of added vectors. In this paper we propose a constructive method that allows to extend a given frame to an oblique dual frame pair such that the number of added vectors is in general much smaller than the number of added vectors used to extend it to a tight frame. We also present a uniqueness characterization and several equivalent characterizations for an oblique dual frame pair, it turns out that oblique dual frame pair provides more flexibilities than alternate dual frame pair.

scholarly journals Operator Characterizations and Some Properties ofg-Frames on Hilbert Spaces

2013 ◽  
Vol 2013 ◽  
pp. 1-9 ◽  
Author(s):  
Xunxiang Guo
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Orthonormal Basis ◽  
Riesz Bases ◽  
Dual Frames

Given theg-orthonormal basis for Hilbert spaceH, we characterize theg-frames, normalized tightg-frames, andg-Riesz bases in terms of theg-preframe operators. Then we consider the transformations ofg-frames, normalized tightg-frames, andg-Riesz bases, which are induced by operators and characterize them in terms of the operators. Finally, we discuss the sums andg-dual frames ofg-frames by applying the results of characterizations.

scholarly journals Dual finite frames for vector spaces over an arbitrary field with applications

2021 ◽  
pp. 1-36
Author(s):  
Patricia Mariela Morillas
Keyword(s):  
Hilbert Spaces ◽  
Matrix Representation ◽  
Vector Spaces ◽  
Complex Numbers ◽  
Arbitrary Field ◽  
Dual Frames ◽  
Finite Frames ◽  
Finite Dimensional ◽  
The Matrix ◽  
Normed Vector

In the present paper, we study frames for finite-dimensional vector spaces over an arbitrary field. We develop a theory of dual frames in order to obtain and study the different representations of the elements of the vector space provided by a frame. We relate the introduced theory with the classical one of dual frames for Hilbert spaces and apply it to study dual frames for three types of vector spaces: for vector spaces over conjugate closed subfields of the complex numbers (in particular, for cyclotomic fields), for metric vector spaces, and for ultrametric normed vector spaces over complete non-archimedean valued fields. Finally, we consider the matrix representation of operators using dual frames and its application to the solution of operators equations in a Petrov-Galerkin scheme.

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