REVERSED WAVELET FUNCTIONS AND SUBSPACES

2007 ◽  
Vol 05 (05) ◽  
pp. 699-707 ◽  
Author(s):  
NHAN LEVAN ◽  
CARLOS S. KUBRUSLY
Keyword(s):  
Orthonormal Basis ◽  
Orthogonal Decomposition ◽  
Wavelet Theory ◽  
Direct Sum ◽  
Multiplicity One ◽  
Reducing Subspaces ◽  
Infinite Multiplicity ◽  
Countably Infinite ◽  
Bilateral Shift

Let the operators D and T be the dilation-by-2 and translation-by-1 on [Formula: see text], which are both bilateral shifts of infinite multiplicity. If ψ(·) in [Formula: see text] is a wavelet, then {DmTnψ(·)}(m,n)∈ℤ2 is an orthonormal basis for the Hilbert space [Formula: see text] but the reversed set {TnDmψ(·)}(n,m)∈ℤ2 is not. In this paper we investigate the role of the reversed functions TnDmψ(·) in wavelet theory. As a consequence, we exhibit an orthogonal decomposition of [Formula: see text] into T-reducing subspaces upon which part of the bilateral shift T consists of a countably infinite direct sum of bilateral shifts of multiplicity one, which mirrors a well-known decomposition of the bilateral shift D.

scholarly journals Reducing Subspaces of Some Multiplication Operators on the Bergman Space over Polydisk

2015 ◽  
Vol 2015 ◽  
pp. 1-12
Author(s):  
Yanyue Shi ◽  
Na Zhou
Keyword(s):  
Bergman Space ◽  
Multiplication Operators ◽  
Direct Sum ◽  
Reducing Subspace ◽  
Reducing Subspaces

We consider the reducing subspaces ofMzNonAα2(Dk), wherek≥3,zN=z1N1⋯zkNk, andNi≠Njfori≠j. We prove that each reducing subspace ofMzNis a direct sum of some minimal reducing subspaces. We also characterize the minimal reducing subspaces in the cases thatα=0andα∈(-1,+∞)∖Q, respectively. Finally, we give a complete description of minimal reducing subspaces ofMzNonAα2(D3)withα>-1.

scholarly journals Finite Dimensional Approximation to Band Limited White Noise

1967 ◽  
Vol 29 ◽  
pp. 211-216 ◽  
Author(s):  
Takeyuki Hida ◽  
Hisao Nomoto
Keyword(s):  
White Noise ◽  
Point Spectrum ◽  
Unitary Operators ◽  
Finite Dimensional Approximation ◽  
Finite Dimensional ◽  
Period 2 ◽  
Dimensional Approximation ◽  
Band Limited ◽  
Infinite Multiplicity ◽  
Countably Infinite

One of the authors discussed finite dimensional approximations to a white noise and a periodic Brownian motion with period 2 π on the projective limit space of spheres ([2]). The group of unitary operators derived from the periodic white noise has a pure point spectrum which consists of all integers with countably infinite multiplicity. We also have much interest in the investigation of a band limited white noise which is another typical example having quite different spectral type. Indeed, the corresponding group of unitary operators has a continuous spectrum with countably infinite multiplicity.

scholarly journals THE SCHWINGER REPRESENTATION OF A GROUP: CONCEPT AND APPLICATIONS

2006 ◽  
Vol 18 (08) ◽  
pp. 887-912 ◽  
Author(s):  
S. CHATURVEDI ◽  
G. MARMO ◽  
N. MUKUNDA ◽  
R. SIMON ◽  
A. ZAMPINI
Keyword(s):  
Quantum Mechanics ◽  
Lie Group ◽  
Irreducible Representations ◽  
Quantum Mechanical ◽  
Direct Sum ◽  
Pure States ◽  
Group Concept ◽  
Set Up ◽  
Multiplicity Free

The concept of the Schwinger Representation of a finite or compact simple Lie group is set up as a multiplicity-free direct sum of all the unitary irreducible representations of the group. This is abstracted from the properties of the Schwinger oscillator construction for SU(2), and its relevance in several quantum mechanical contexts is highlighted. The Schwinger representations for SU(2), SO(3) and SU(n) for all n are constructed via specific carrier spaces and group actions. In the SU(2) case, connections to the oscillator construction and to Majorana's theorem on pure states for any spin are worked out. The role of the Schwinger Representation in setting up the Wigner–Weyl isomorphism for quantum mechanics on a compact simple Lie group is brought out.

Semi-Orthogonal Parseval Wavelets Frames on Local Fields and Applications in Manufacturing Science

2013 ◽  
Vol 712-715 ◽  
pp. 2464-2468
Author(s):  
Shi Heng Wang
Keyword(s):  
Local Fields ◽  
Affine Subspace ◽  
Parseval Frame ◽  
Direct Sum ◽  
Reducing Subspaces ◽  
Dilation Factor ◽  
Definition Of ◽  
Frame Wavelets ◽  
Manufacturing Science ◽  
Orthogonal Frame

Manufacturing science focuses on understanding problems from the perspective of the stakeholders involved and then applying manufacturing science as needed. We investigate semi-orthogonal frame wavelets and Parseval frame wavelets in with a dilation factor. We show that every affine subspace is the orthogonal direct sum of at most three purely non-reducing subspaces. This result is obtained through considering the basicquestion as to when the orthogonal complement of an afffine subspace in another one is still affine subspace.The definition of multiple pseudofames for subspaces with integer translation is proposed. The notion of a generalized multiresolution structure of is also introduced. The construction of a generalized multireso-lution structure of Paley-Wiener subspaces of is investigated.

The Structure and Properties of a Class of Affine Subspaces and Applications in Mechatronics Science

2013 ◽  
Vol 321-324 ◽  
pp. 2385-2388
Author(s):  
Su Luo Song
Keyword(s):  
Information Science ◽  
Fundamental Question ◽  
Orthogonal Complement ◽  
Affine Subspace ◽  
Wavelet Frame ◽  
Structure And Properties ◽  
Direct Sum ◽  
Affine Subspaces ◽  
Reducing Subspaces

Information science focuses on understanding problems from the perspective of the stakeholders involved and then applying information and other technologies as needed. We show that every affine subspace is the orthogonal direct sum of at most three purely non-reducing subspaces. This result is obtained through considering the basicquestion as to when the orthogonal complement of an afffine subspace in another one is still affine subspace. Motivated by the fundamental question as to whethor every affine subspace is singly-generated wavelet frame, we prove that every affine sub -space can be decomposed into the direct sum of a singly-generated afffine subspace.

DUAL-SHIFT DECOMPOSITION AND WAVELETS

2014 ◽  
Vol 12 (02) ◽  
pp. 1450014
Author(s):  
NHAN LEVAN ◽  
CARLOS S. KUBRUSLY
Keyword(s):  
Hilbert Space ◽  
Multiresolution Analysis ◽  
Orthonormal Basis ◽  
Haar System ◽  
Wavelet Theory ◽  
System Type

We introduce the notion of dual-shift decomposition of an arbitrary Hilbert space, which is given in terms of two unilateral shifts. After ensuring conditions for the existence of it, such a decomposition is then constructed for the concrete space ℒ2[0, 1], on which the two unilateral shifts are parts of the dilation-by-2 and the translation-by-1 on ℒ2(ℝ). Using multiresolution analysis (MRA) of wavelet theory it is shown the existence of a Haar-system-type orthonormal basis for ℒ2[0, 1], which is combined with the dual-shift decomposition to yield a refined decomposition for ℒ2[0, 1].

WAVELET-BASED FEATURE EXTRACTION FOR MORTALITY PROJECTION

2020 ◽  
Vol 50 (3) ◽  
pp. 675-707
Author(s):  
Donatien Hainaut ◽  
Michel Denuit
Keyword(s):  
Orthonormal Basis ◽  
Mortality Rates ◽  
Wavelet Theory ◽  
Chi Square ◽  
Mortality Forecasting ◽  
Mortality Projection ◽  
Chi Square Test ◽  
Belgian Population ◽  
Likelihood Approach ◽  
Penalized Likelihood Approach

AbstractWavelet theory is known to be a powerful tool for compressing and processing time series or images. It consists in projecting a signal on an orthonormal basis of functions that are chosen in order to provide a sparse representation of the data. The first part of this article focuses on smoothing mortality curves by wavelets shrinkage. A chi-square test and a penalized likelihood approach are applied to determine the optimal degree of smoothing. The second part of this article is devoted to mortality forecasting. Wavelet coefficients exhibit clear trends for the Belgian population from 1965 to 2015, they are easy to forecast resulting in predicted future mortality rates. The wavelet-based approach is then compared with some popular actuarial models of Lee–Carter type estimated fitted to Belgian, UK, and US populations. The wavelet model outperforms all of them.

scholarly journals Energy and Entropy in Turbulence Decompositions

Entropy ◽  
2019 ◽  
Vol 21 (2) ◽  
pp. 124
Author(s):  
Václav Uruba
Keyword(s):  
Decomposition Methods ◽  
Orthogonal Decomposition ◽  
Efficiency Evaluation ◽  
Reduced Order ◽  
Energy Concept ◽  
Velocity Flow ◽  
Definition Of ◽  
Proper Orthogonal ◽  
Specific Definition

The role of energy and entropy in the decomposition of turbulent velocity flow-fields is shown in this paper. Decomposition methods based on the energy concept are taken into account—proper orthogonal decomposition (POD) and its extension bi-orthogonal decomposition (BOD). The methods are well known; however, various versions are used and the interpretation of results is not straightforward. To make this clearer, the specific definition of modes is suggested and specified; moreover, energy- and entropy-motivated views on the decomposed modes are presented. This concept could offer new possibilities in the physical interpretation of modes and in reduced-order modeling (ROM) strategy efficiency evaluation.

scholarly journals Structure of Iso-Symmetric Operators

Axioms ◽  
2021 ◽  
Vol 10 (4) ◽  
pp. 256
Author(s):  
Bhagwati Prashad Duggal ◽  
In-Hyoun Kim
Keyword(s):  
Matrix Representation ◽  
Triangular Matrix ◽  
Space Operator ◽  
Left Multiplication ◽  
Direct Sum ◽  
Upper Triangular Matrix ◽  
Symmetric Operators ◽  
Bilateral Shift ◽  
Positive Integers ◽  
Power Bounded

For a Hilbert space operator T∈B(H), let LT and RT∈B(B(H)) denote, respectively, the operators of left multiplication and right multiplication by T. For positive integers m and n, let ▵T∗,Tm(I)=(LT∗RT−I)m(I) and δT∗,Tn(I)=(LT∗−RT)m(I). The operator T is said to be (m,n)-isosymmetric if ▵T∗,TmδT∗,Tn(I)=0. Power bounded (m,n)-isosymmetric operators T∈B(H) have an upper triangular matrix representation T=T1T30T2∈B(H1⊕H2) such that T1∈B(H1) is a C0.-operator which satisfies δT1∗,T1n(I|H1)=0 and T2∈B(H2) is a C1.-operator which satisfies AT2=(Vu⊕Vb)|H2A, A=limt→∞T2∗tT2t, Vu is a unitary and Vb is a bilateral shift. If, in particular, T is cohyponormal, then T is the direct sum of a unitary with a C00-contraction.

scholarly journals Energy and Entropy in Turbulence Decompositions

2018 ◽  
Author(s):  
Václav Uruba
Keyword(s):  
Proper Orthogonal Decomposition ◽  
Physical Interpretation ◽  
Decomposition Methods ◽  
Orthogonal Decomposition ◽  
Efficiency Evaluation ◽  
Reduced Order ◽  
Energy Concept ◽  
Velocity Flow ◽  
Proper Orthogonal

The role of energy and entropy in decomposition of turbulent velocity flow-fields is to be shown in the paper. The decomposition methods based on energy concept are taken into account, namely the Proper Orthogonal Decomposition (POD) and its extension Bi-Orthogonal Decomposition (BOD). Entropy motivated view on the decomposed modes could offer new possibilities in the modes physical interpretation and in Reduced Order Modelling (ROM) strategy efficiency evaluation.

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