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 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 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.

DISCRETE-TIME WILSON FRAMES WITH GENERAL LATTICES

2012 ◽  
Vol 10 (05) ◽  
pp. 1250044 ◽  
Author(s):  
QIAOFANG LIAN ◽  
YUNFANG LIAN ◽  
MINGHOU YOU
Keyword(s):  
Discrete Time ◽  
Tight Frame ◽  
Gabor Frame ◽  
Dual Frame ◽  
Necessary And Sufficient Condition ◽  
Dual Frames ◽  
Gabor Dual ◽  
Bessel Sequences ◽  
Necessary And Sufficient

In this paper, we focus on the construction of Wilson frames and their dual frames for general lattices of volume [Formula: see text] (K even) in the discrete-time setting. We obtain a necessary and sufficient condition for two Bessel sequences having Wilson structure to be dual frames for l2(ℤ). When the window function satisfies some symmetry property, we obtain a characterization of a Wilson system to be a tight frame for l2(ℤ), show that a Wilson frame for l2(ℤ) can be derived from the underlying Gabor frame, and that the dual frame having Wilson structure can also be derived from the canonical Gabor dual of the underlying Gabor frame.

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.

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.

Wilson frames for ℂL with general lattices

2016 ◽  
Vol 14 (06) ◽  
pp. 1650055
Author(s):  
Minghou You ◽  
Junqiao Yang ◽  
Qiaofang Lian
Keyword(s):  
Digital Signal ◽  
Tight Frame ◽  
Gabor Frame ◽  
Dual Frame ◽  
Dual Frames ◽  
Volume Formula ◽  
Discrete Signals ◽  
Gabor Dual ◽  
Necessary And Sufficient

In digital signal and image processing one can only process discrete signals of finite length, and the space [Formula: see text] is the preferred setting. Recently, Kutyniok and Strohmer constructed orthonormal Wilson bases for [Formula: see text] with general lattices of volume [Formula: see text] ([Formula: see text] even). In this paper, we extend this construction to Wilson frames for [Formula: see text] with general lattices of volume [Formula: see text], where [Formula: see text] and [Formula: see text]. We obtain a necessary and sufficient condition for two sequences having Wilson structure to be dual frames for [Formula: see text]. When the window function satisfies some symmetry property, we obtain a characterization of a Wilson system to be a tight frame for [Formula: see text], show that a Wilson frame for [Formula: see text] can be derived from the underlying Gabor frame, and that the dual frame having Wilson structure can also be derived from the canonical Gabor dual of the underlying Gabor frame.

Linear Algebra and the Dirac Notation

2006 ◽  
Author(s):  
Phillip Kaye ◽  
Raymond Laflamme ◽  
Michele Mosca
Keyword(s):  
Hilbert Space ◽  
Quantum Mechanics ◽  
Quantum Computing ◽  
Linear Algebra ◽  
Hilbert Spaces ◽  
Vector Spaces ◽  
Complex Vector ◽  
Dual Vector ◽  
Finite Dimensional ◽  
Basic Facts

We assume the reader has a strong background in elementary linear algebra. In this section we familiarize the reader with the algebraic notation used in quantum mechanics, remind the reader of some basic facts about complex vector spaces, and introduce some notions that might not have been covered in an elementary linear algebra course. The linear algebra notation used in quantum computing will likely be familiar to the student of physics, but may be alien to a student of mathematics or computer science. It is the Dirac notation, which was invented by Paul Dirac and which is used often in quantum mechanics. In mathematics and physics textbooks, vectors are often distinguished from scalars by writing an arrow over the identifying symbol: e.g a⃗. Sometimes boldface is used for this purpose: e.g. a. In the Dirac notation, the symbol identifying a vector is written inside a ‘ket’, and looks like |a⟩. We denote the dual vector for a (defined later) with a ‘bra’, written as ⟨a|. Then inner products will be written as ‘bra-kets’ (e.g. ⟨a|b⟩). We now carefully review the definitions of the main algebraic objects of interest, using the Dirac notation. The vector spaces we consider will be over the complex numbers, and are finite-dimensional, which significantly simplifies the mathematics we need. Such vector spaces are members of a class of vector spaces called Hilbert spaces. Nothing substantial is gained at this point by defining rigorously what a Hilbert space is, but virtually all the quantum computing literature refers to a finite-dimensional complex vector space by the name ‘Hilbert space’, and so we will follow this convention. We will use H to denote such a space. Since H is finite-dimensional, we can choose a basis and alternatively represent vectors (kets) in this basis as finite column vectors, and represent operators with finite matrices. As you see in Section 3, the Hilbert spaces of interest for quantum computing will typically have dimension 2n, for some positive integer n. This is because, as with classical information, we will construct larger state spaces by concatenating a string of smaller systems, usually of size two.

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