Duality relations for a class of generalized weak R-duals

2021 ◽  
pp. 2050085
Author(s):  
Jian Dong ◽  
Yun-Zhang Li
Keyword(s):  
Frame Theory ◽  
Riesz Bases ◽  
Duality Principle ◽  
General Version ◽  
Gabor Analysis ◽  
General Frame ◽  
Duality Relations

Since the introduction of R-duals by Casazza, Kutyniok and Lammers with the motivation to obtain a general version of duality principle in Gabor analysis, various R-duals and some relaxations of the R-dual setup have been introduced and studied by some mathematicians. They provide a powerful tool in the analysis of duality relations in general frame theory, and are far beyond the duality principle in Gabor analysis. In this paper, we introduce the concept of generalized weak R-dual based on a pair of frames which is a relaxation of the R-dual setup. Using generalized weak R-duals, we characterize the frame properties of a sequence and the equivalence between two frames, prove that the generalized weak R-duals of frames (Riesz bases) are frame sequences (frames), and present a coefficient expression corresponding to the canonical duals of generalized weak R-duals. Some examples are provided to illustrate the generality of the theory.

Bessel Sequences and Its F-Scalability

2015 ◽  
Vol 7 (4) ◽  
pp. 441-453 ◽  
Author(s):  
Lei Liu ◽  
Xianwei Zheng ◽  
Jingwen Yan ◽  
Xiaodong Niu
Keyword(s):  
Quantum Mechanics ◽  
Wavelet Analysis ◽  
Frame Theory ◽  
Riesz Bases ◽  
Gabor Analysis ◽  
Perturbation Result ◽  
Extension Principles ◽  
Bessel Sequences

AbstractFrame theory, which contains wavelet analysis and Gabor analysis, has become a powerful tool for many applications of mathematics, engineering and quantum mechanics. The study of extension principles of Bessel sequences to frames is important in frame theory. This paper studies transformations on Bessel sequences to generate frames and Riesz bases in terms of operators and scalability. Some characterizations of operators that mapping Bessel sequences to frames and Riesz bases are given. We introduce the definitions of F-scalable and P-scalable Bessel sequences. F-scalability and P-scalability of Bessel sequences are discussed in this paper, then characterizations of scalings of F-scalable or P-scalable Bessel sequences are established. Finally, a perturbation result on F-scalable Bessel sequences is derived.

scholarly journals A Duality Principle for Groups II: Multi-frames Meet Super-Frames

2020 ◽  
Vol 26 (6) ◽  
Author(s):  
R. Balan ◽  
D. Dutkay ◽  
D. Han ◽  
D. Larson ◽  
F. Luef
Keyword(s):  
Group Representation ◽  
Duality Principle ◽  
Group Representations ◽  
Fundamental Identity ◽  
Gabor Analysis ◽  
Time Frequency ◽  
Fundamental Properties ◽  
Super Frame ◽  
Funct Anal ◽  
General Duality

AbstractThe duality principle for group representations developed in Dutkay et al. (J Funct Anal 257:1133–1143, 2009), Han and Larson (Bull Lond Math Soc 40:685–695, 2008) exhibits a fact that the well-known duality principle in Gabor analysis is not an isolated incident but a more general phenomenon residing in the context of group representation theory. There are two other well-known fundamental properties in Gabor analysis: the biorthogonality and the fundamental identity of Gabor analysis. The main purpose of this this paper is to show that these two fundamental properties remain to be true for general projective unitary group representations. Moreover, we also present a general duality theorem which shows that that muti-frame generators meet super-frame generators through a dual commutant pair of group representations. Applying it to the Gabor representations, we obtain that $$\{\pi _{\Lambda }(m, n)g_{1} \oplus \cdots \oplus \pi _{\Lambda }(m, n)g_{k}\}_{m, n \in {\mathbb {Z}}^{d}}$$ { π Λ ( m , n ) g 1 ⊕ ⋯ ⊕ π Λ ( m , n ) g k } m , n ∈ Z d is a frame for $$L^{2}({\mathbb {R}}\,^{d})\oplus \cdots \oplus L^{2}({\mathbb {R}}\,^{d})$$ L 2 ( R d ) ⊕ ⋯ ⊕ L 2 ( R d ) if and only if $$\cup _{i=1}^{k}\{\pi _{\Lambda ^{o}}(m, n)g_{i}\}_{m, n\in {\mathbb {Z}}^{d}}$$ ∪ i = 1 k { π Λ o ( m , n ) g i } m , n ∈ Z d is a Riesz sequence, and $$\cup _{i=1}^{k} \{\pi _{\Lambda }(m, n)g_{i}\}_{m, n\in {\mathbb {Z}}^{d}}$$ ∪ i = 1 k { π Λ ( m , n ) g i } m , n ∈ Z d is a frame for $$L^{2}({\mathbb {R}}\,^{d})$$ L 2 ( R d ) if and only if $$\{\pi _{\Lambda ^{o}}(m, n)g_{1} \oplus \cdots \oplus \pi _{\Lambda ^{o}}(m, n)g_{k}\}_{m, n \in {\mathbb {Z}}^{d}}$$ { π Λ o ( m , n ) g 1 ⊕ ⋯ ⊕ π Λ o ( m , n ) g k } m , n ∈ Z d is a Riesz sequence, where $$\pi _{\Lambda }$$ π Λ and $$\pi _{\Lambda ^{o}}$$ π Λ o is a pair of Gabor representations restricted to a time–frequency lattice $$\Lambda $$ Λ and its adjoint lattice $$\Lambda ^{o}$$ Λ o in $${\mathbb {R}}\,^{d}\times {\mathbb {R}}\,^{d}$$ R d × R d .

scholarly journals Gabor duality theory for Morita equivalent C∗-algebras

2020 ◽  
Vol 31 (10) ◽  
pp. 2050073 ◽  
Author(s):  
Are Austad ◽  
Mads S. Jakobsen ◽  
Franz Luef
Keyword(s):  
Duality Theory ◽  
Morita Equivalence ◽  
Gabor Frames ◽  
Duality Principle ◽  
Gabor Analysis ◽  
Matrix Algebras ◽  
C Algebras ◽  
Morita Equivalent ◽  
Twisted Group ◽  
Density Theorems

The duality principle for Gabor frames is one of the pillars of Gabor analysis. We establish a far-reaching generalization to Morita equivalence bimodules with some extra properties. For certain twisted group [Formula: see text]-algebras, the reformulation of the duality principle to the setting of Morita equivalence bimodules reduces to the well-known Gabor duality principle by localizing with respect to a trace. We may lift all results at the module level to matrix algebras and matrix modules, and in doing so, it is natural to introduce [Formula: see text]-matrix Gabor frames, which generalize multi-window super Gabor frames. We are also able to establish density theorems for module frames on equivalence bimodules, and these localize to density theorems for [Formula: see text]-matrix Gabor frames.

Signal Representations Via SIP p-frames and SIP Bessel Multipliers in Separable Banach Spaces

2021 ◽  
pp. 2150005
Author(s):  
Xianwei Zheng ◽  
Cuiming Zou ◽  
Shouzhi Yang
Keyword(s):  
Banach Spaces ◽  
Frame Theory ◽  
Inner Product ◽  
Riesz Bases ◽  
Digital Signals ◽  
Bessel Sequence ◽  
Fixed Sequence ◽  
Signal Representations ◽  
Special Cases ◽  
Bessel Multipliers

Digital signals are often modeled as functions in Banach spaces, such as the ubiquitous [Formula: see text] spaces. The frame theory in Banach spaces induces flexible representations of signals due to the robustness and redundancy of frames. Nevertheless, the lack of inner product in general Banach spaces limits the direct representations of signals in Banach spaces under a given basis or frame. In this paper, we introduce the concept of semi-inner product (SIP) [Formula: see text]-Bessel multipliers to extend the flexibility of signal representations in separable Banach spaces, where [Formula: see text]. These multipliers are defined as composition of analysis operator of an SIP-I Bessel sequence, a multiplication with a fixed sequence and synthesis operator of an SIP-II Bessel sequence. The basic properties of the SIP [Formula: see text]-Bessel multipliers are investigated. Moreover, as special cases, characterizations of [Formula: see text]-Riesz bases related to signal representations are given, and the multipliers for [Formula: see text]-Riesz bases are discussed. We show that SIP [Formula: see text]-Bessel multipliers for [Formula: see text]-Riesz bases are invertible. Finally, the continuity of SIP [Formula: see text]-Bessel multipliers with respect to their parameters is investigated. The results theoretically show that the SIP [Formula: see text]-Bessel multipliers offer a larger range of freedom than frames on signal representations in Banach spaces.

scholarly journals KACZMARZ ALGORITHM AND FRAMES

2013 ◽  
Vol 11 (05) ◽  
pp. 1350036 ◽  
Author(s):  
WOJCIECH CZAJA ◽  
JAMES H. TANIS
Keyword(s):  
Infinite System ◽  
Frame Theory ◽  
Riesz Bases ◽  
Tight Frames ◽  
Algebraic Equations ◽  
Orthonormal Bases ◽  
Kaczmarz Algorithm ◽  
Linear Algebraic Equations ◽  
Bounded Matrices ◽  
Unit Vectors

Sequences of unit vectors for which the Kaczmarz algorithm always converges in Hilbert space can be characterized in frame theory by tight frames with constant 1. We generalize this result to the context of frames and bases. In particular, we show that the only effective sequences which are Riesz bases are orthonormal bases. Moreover, we consider the infinite system of linear algebraic equations Ax = b and characterize the (bounded) matrices A for which the Kaczmarz algorithm always converges to a solution.

scholarly journals Wavelet bases for a unitary operator

1995 ◽  
Vol 38 (2) ◽  
pp. 233-260 ◽  
Author(s):  
S. L. Lee ◽  
H. H. Tan ◽  
W. S. Tang
Keyword(s):  
Hilbert Space ◽  
Riesz Basis ◽  
Unitary Operator ◽  
Linear Span ◽  
General Setting ◽  
Complex Hilbert Space ◽  
Riesz Bases ◽  
Duality Principle ◽  
Spline Wavelets ◽  
Necessary And Sufficient

Let T be a unitary operator on a complex Hilbert space ℋ, and X, Y be finite subsets of ℋ. We give a necessary and sufficient condition for TZ(X): {Tnx: n ∈ Z, x ∈ X} to be a Riesz basis of its closed linear span 〈TZ(X)〉. If TZ(X) and TZ(Y) are Riesz bases, and 〈TZ(X)〉⊂〈TZ(Y)〉, then X is extendable to X′ such that TZ(X′) is a Riesz basis of TZ(Y) The proof provides an algorithm for the construction of Riesz bases for the orthogonal complement of 〈TZ(X)〉 in 〈TZ(Y)〉. In the case X consists of a single B-spline, the algorithm gives a natural and quick construction of the spline wavelets of Chui and Wang [2, 3]. Further, the duality principle of Chui and Wang in [3] and [4] is put in the general setting of biorthogonal Riesz bases in Hilbert space.

scholarly journals Discrete Subspace Multiwindow Gabor Frames and Their Duals

2013 ◽  
Vol 2013 ◽  
pp. 1-17 ◽  
Author(s):  
Yun-Zhang Li ◽  
Yan Zhang
Keyword(s):  
Gabor Frames ◽  
Riesz Bases ◽  
Type I ◽  
Type Ii ◽  
Zak Transform ◽  
Gabor Analysis ◽  
Time Frequency ◽  
Orthonormal Bases ◽  
Gabor Dual ◽  
Discrete Subspace

This paper addresses discrete subspace multiwindow Gabor analysis. Such a scenario can model many practical signals and has potential applications in signal processing. In this paper, using a suitable Zak transform matrix we characterize discrete subspace mixed multi-window Gabor frames (Riesz bases and orthonormal bases) and their duals with Gabor structure. From this characterization, we can easily obtain frames by designing Zak transform matrices. In particular, for usual multi-window Gabor frames (i.e., all windows have the same time-frequency shifts), we characterize the uniqueness of Gabor dual of type I (type II) and also give a class of examples of Gabor frames and an explicit expression of their Gabor duals of type I (type II).

scholarly journals On R-Duals and the Duality Principle in Gabor Analysis

2014 ◽  
Vol 21 (2) ◽  
pp. 383-400 ◽  
Author(s):  
Diana T. Stoeva ◽  
Ole Christensen
Keyword(s):  
Duality Principle ◽  
Gabor Analysis
Sign in / Sign up

Export Citation Format

Share Document