scholarly journals Orthonormal bases and quasi-splitting subspaces in pre-Hilbert spaces

2008 ◽  
Vol 345 (2) ◽  
pp. 725-730 ◽  
Author(s):  
D. Buhagiar ◽  
E. Chetcuti ◽  
H. Weber
Keyword(s):  
Hilbert Spaces ◽  
Orthonormal Bases

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 Quantum ergodicity of random orthonormal bases of spaces of high dimension

2014 ◽  
Vol 372 (2007) ◽  
pp. 20120511 ◽  
Author(s):  
Steve Zelditch
Keyword(s):  
Hilbert Spaces ◽  
Orthonormal Basis ◽  
Compact Riemannian Manifold ◽  
Random Sequence ◽  
Limit State ◽  
Flat Torus ◽  
Quantum Ergodicity ◽  
Orthonormal Bases ◽  
Finite Dimensional ◽  
Unique Limit

We consider a sequence of finite-dimensional Hilbert spaces of dimensions . Motivating examples are eigenspaces, or spaces of quasi-modes, for a Laplace or Schrödinger operator on a compact Riemannian manifold. The set of Hermitian orthonormal bases of may be identified with U ( d N ), and a random orthonormal basis of is a choice of a random sequence U N ∈ U ( d N ) from the product of normalized Haar measures. We prove that if and if tends to a unique limit state ω ( A ), then almost surely an orthonormal basis is quantum ergodic with limit state ω ( A ). This generalizes an earlier result of the author in the case where is the space of spherical harmonics on S 2 . In particular, it holds on the flat torus if d ≥5 and shows that a highly localized orthonormal basis can be synthesized from quantum ergodic ones and vice versa in relatively small dimensions.

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.

scholarly journals On a Class of Self-Adjoint Compact Operators in Hilbert Spaces and Their Relations with Their Finite-Range Truncations

2013 ◽  
Vol 2013 ◽  
pp. 1-14
Author(s):  
M. De la Sen
Keyword(s):  
Fixed Points ◽  
Hilbert Spaces ◽  
Compact Operators ◽  
Finite Range ◽  
Worst Case ◽  
Orthonormal Bases ◽  
Finite Dimensional

This paper investigates a class of self-adjoint compact operators in Hilbert spaces related to their truncated versions with finite-dimensional ranges. The comparisons are established in terms of worst-case norm errors of the composite operators generated from iterated computations. Some boundedness properties of the worst-case norms of the errors in their respective fixed points in which they exist are also given. The iterated sequences are expanded in separable Hilbert spaces through the use of numerable orthonormal bases.

scholarly journals The Singular Value Expansion for Arbitrary Bounded Linear Operators

Mathematics ◽  
2020 ◽  
Vol 8 (8) ◽  
pp. 1346
Author(s):  
Daniel K. Crane ◽  
Mark S. Gockenbach
Keyword(s):  
Linear Operator ◽  
Hilbert Spaces ◽  
Singular Value ◽  
Linear Operators ◽  
Simple Analysis ◽  
Basic Tool ◽  
Bounded Linear ◽  
Orthonormal Bases ◽  
Value Decomposition ◽  
Well Posed

The singular value decomposition (SVD) is a basic tool for analyzing matrices. Regarding a general matrix as defining a linear operator and choosing appropriate orthonormal bases for the domain and co-domain allows the operator to be represented as multiplication by a diagonal matrix. It is well known that the SVD extends naturally to a compact linear operator mapping one Hilbert space to another; the resulting representation is known as the singular value expansion (SVE). It is less well known that a general bounded linear operator defined on Hilbert spaces also has a singular value expansion. This SVE allows a simple analysis of a variety of questions about the operator, such as whether it defines a well-posed linear operator equation and how to regularize the equation when it is not well posed.

scholarly journals g-Bases in Hilbert Spaces

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Xunxiang Guo
Keyword(s):  
Hilbert Spaces ◽  
Schauder Basis ◽  
Orthonormal Bases ◽  
Schauder Bases

The concept ofg-basis in Hilbert spaces is introduced, which generalizes Schauder basis in Hilbert spaces. Some results aboutg-bases are proved. In particular, we characterize theg-bases andg-orthonormal bases. And the dualg-bases are also discussed. We also consider the equivalent relations ofg-bases andg-orthonormal bases. And the property ofg-minimal ofg-bases is studied as well. Our results show that, in some cases,g-bases share many useful properties of Schauder bases in Hilbert spaces.

Homeomorphic Images of Orthogonal Bases

2016 ◽  
Vol 31 ◽  
pp. 485-491
Author(s):  
M. Kebryaee ◽  
M. Radjabalipour
Keyword(s):  
Hilbert Space ◽  
Linear Algebra ◽  
Hilbert Spaces ◽  
Finite Subset ◽  
Sufficient Conditions ◽  
Orthogonal Basis ◽  
Riesz Bases ◽  
Orthonormal Bases ◽  
Homeomorphic Image ◽  
Necessary And Sufficient

Necessary and sufficient conditions are obtained for a sequence $\{x_j:~j\in \mathbb J\}$ in a Hilbert space to be, up to the elimination of a finite subset of $\mathbb J$, the linear homeomorphic image of an orthogonal basis of some Hilbert space $K$. This extends a similar result for orthonormal bases due to Holub [J.R. Holub. Pre-frame operators, Besselian frames, and near-Riesz bases in Hilbert spaces. \textit{Proc. Amer. Math. Soc.}, 122(3):779--785, 1994]. The proofs given here are based on simple linear algebra techniques.

scholarly journals Hilbert modules revisited: orthonormal bases and Hilbert-Schmidt operators

1995 ◽  
Vol 37 (1) ◽  
pp. 45-54
Author(s):  
M. Cabrera ◽  
J. Martínez ◽  
A. Rodríguez
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Complex Hilbert Space ◽  
Hilbert Module ◽  
Structure Theory ◽  
Complex Field ◽  
Hilbert Modules ◽  
Complete Structure ◽  
Orthonormal Bases

The concept of a Hilbert module (over an H*-algebra) arises as a generalization of that of a complex Hilbert space when the complex field is replaced by an (associative) H*-algebra with zero annihilator. P. P. Saworotnow [13] introduced Hilbert modules and extended to its context some classical theorems from the theory of Hilbert spaces, J. F. Smith [17] gave a complete structure theory for Hilbert modules, and G. R. Giellis [9] obtained a nice characteristization of Hilbert modules.

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