Oblique dual frames in Hilbert spaces

2019 ◽  
Vol 13 (06) ◽  
pp. 2050117
Author(s):  
Hossein Javanshiri ◽  
Abdolmajid Fattahi ◽  
Mojtaba Sargazi
Keyword(s):  
Hilbert Spaces ◽  
Surprising Result ◽  
Dual Frames ◽  
Frame Sequence ◽  
Oblique Dual

In this paper, we consider oblique dual frames and particularly, we obtain some of their characterizations. Among other things, special attention is devoted to the study of the effect of perturbations of frame sequences on their oblique duals. In particular, as a surprising result, we show that a frame sequence is uniquely determined by the set of its oblique dual.

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.

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.

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.

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.

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.

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