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].

Biorthogonal Systems in lp-Spaces

1969 ◽  
Vol 21 ◽  
pp. 625-638 ◽  
Author(s):  
R. Keown ◽  
C. Conatser
Keyword(s):  
Hilbert Space ◽  
Hilbert Spaces ◽  
Orthonormal Basis ◽  
Natural Generalization ◽  
Standard Basis ◽  
Biorthogonal Systems ◽  
Lp Spaces ◽  
Standard Bases

Our aim in this paper is to generalize certain ideas and results of Bary (1) on biorthogonal systems in separable Hilbert spaces to their counterparts in separable lp-spaces, 1 < p.The main result of Bary is to characterize a natural generalization of the concept of orthonormal basis for a Hilbert space. That of this paper is to characterize the concept of a Bary basis which is a generalization of the idea of standard basis of an lp-space. The result is interesting for lp-spaces because of the paucity of standard bases in these spaces.Before summarizing our results, we shall introduce some notation and recall a few pertinent definitions and facts. The symbols and denote mutually conjugate lp-spaces, where is the space lt and the space lswith 1 < r <2 and 2 < s = r/(r – 1).

SYMPLECTIC SPACE OR ORTHOGONAL SPACE OF n QUBITS

2011 ◽  
Vol 09 (06) ◽  
pp. 1449-1457
Author(s):  
JIAN-WEI XU
Keyword(s):  
Hilbert Space ◽  
Orthogonal Group ◽  
Symplectic Group ◽  
Orthonormal Basis ◽  
Mathematical Structure ◽  
Symplectic Space ◽  
Orthogonal Space ◽  
Spin Flip

In Hilbert space of n qubits, we introduce symplectic space (n odd) or orthogonal space (n even) via the spin-flip operator. Under this mathematical structure we discuss some properties of n qubits, including homomorphically mapping local operations of n qubits into symplectic group or orthogonal group, and proving that the generalized "magic basis" is just the biorthonormal basis (i.e. the orthonormal basis of both Hilbert space and the orthogonal space). Finally, a demonstrated example is given to discuss the application in physics of this mathematical structure.

Involutory *-antiautomorphisms in Toeplitz algebras

1988 ◽  
Vol 103 (3) ◽  
pp. 473-480
Author(s):  
P. J. Stacey
Keyword(s):  
Hilbert Space ◽  
Orthonormal Basis ◽  
Compact Operators ◽  
Complex Hilbert Space ◽  
Unilateral Shift ◽  
Integral Power

Let H be a separable complex Hilbert space with orthonormal basis {ei: i ∈ ℕ}, let s be the unilateral shift defined by sei = ei+1 for each i and let K be the algebra of compact operators on H. The present paper classifies the involutory *-anti-automorphisms in the C*-algebra C*(sn, K) generated by K and a positive integral power sn of s. It is shown that, up to conjugacy by *-automorphisms, there are two such involutory *-antiautomorphisms when n is even and one when n is odd.

Vector-valued nonuniform multiresolution analysis on local fields

2015 ◽  
Vol 13 (04) ◽  
pp. 1550029 ◽  
Author(s):  
Firdous Ahmad Shah ◽  
M. Younus Bhat
Keyword(s):  
Multiresolution Analysis ◽  
Orthonormal Basis ◽  
Positive Characteristic ◽  
Local Fields ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Vilenkin Groups ◽  
Refinement Mask ◽  
Necessary And Sufficient ◽  
Vector Valued

A multiresolution analysis (MRA) on local fields of positive characteristic was defined by Shah and Abdullah for which the translation set is a discrete set which is not a group. In this paper, we continue the study based on this nonstandard setting and introduce vector-valued nonuniform multiresolution analysis (VNUMRA) where the associated subspace V0 of L2(K, ℂM) has an orthonormal basis of the form {Φ (x - λ)}λ∈Λ where Λ = {0, r/N} + 𝒵, N ≥ 1 is an integer and r is an odd integer such that r and N are relatively prime and 𝒵 = {u(n) : n ∈ ℕ0}. We establish a necessary and sufficient condition for the existence of associated wavelets and derive an algorithm for the construction of VNUMRA on local fields starting from a vector refinement mask G(ξ) with appropriate conditions. Further, these results also hold for Cantor and Vilenkin groups.

scholarly journals On normal derivations of Hilbert–Schmidt type

1987 ◽  
Vol 29 (2) ◽  
pp. 245-248 ◽  
Author(s):  
Fuad Kittaneh
Keyword(s):  
Hilbert Space ◽  
Orthonormal Basis ◽  
Trace Class ◽  
Linear Operators ◽  
Inner Product ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Infinite Dimensional Hilbert Space ◽  
Schmidt Norm ◽  
Image Position

Let H denote a separable, infinite dimensional Hilbert space. Let B(H), C2 and C1 denote the algebra of all bounded linear operators acting on H, the Hilbert–Schmidt class and the trace class in B(H) respectively. It is well known that C2 and C1 each form a two-sided-ideal in B(H) and C2 is itself a Hilbert space with the inner productwhere {ei} is any orthonormal basis of H and tr(.) is the natural trace on C1. The Hilbert–Schmidt norm of X ∈ C2 is given by ⅡXⅡ2=(X, X)½.

Completeness properties in L2 of the eigenfunctions of two semi-linear differential operators

1980 ◽  
Vol 88 (3) ◽  
pp. 451-468 ◽  
Author(s):  
L. E. Fraenkel
Keyword(s):  
Hilbert Space ◽  
Product Space ◽  
Orthonormal Basis ◽  
Differential Operators ◽  
Real Hilbert Space ◽  
The Real ◽  
Linear Differential Operators ◽  
Linear Problems ◽  
Linear Differential ◽  
Image Position

This paper concerns the boundary-value problemsin which λ is a real parameter, u is to be a real-valued function in C2[0, 1], and problem (I) is that with the minus sign. (The differential operators are called semi-linear because the non-linearity is only in undifferentiated terms.) If we linearize the equations (for ‘ small’ solutions u) by neglecting , there result the eigenvalues λ = n2π2 (with n = 1,2,…) and corresponding normalized eigenfunctionsand it is well known ((2), p. 186) that the sequence {en} is complete in that it is an orthonormal basis for the real Hilbert space L2(0, 1). We shall be concerned with possible extensions of this result to the non-linear problems (I) and (II), for which non-trivial solutions (λ, u) bifurcate from the trivial solution (λ, 0) at the points {n2π2,0) in the product space × L2(0, 1). (Here denotes the real line.)

scholarly journals Quaternionic Blaschke Group

Mathematics ◽  
2018 ◽  
Vol 7 (1) ◽  
pp. 33
Author(s):  
Margit Pap ◽  
Ferenc Schipp
Keyword(s):  
Analytic Function ◽  
Multiresolution Analysis ◽  
Function Spaces ◽  
Complex Case ◽  
Wavelet Theory ◽  
Variable Function ◽  
Analytic Function Spaces

In the complex case, the Blaschke group was introduced and studied. It turned out that in the complex case this group plays important role in the construction of analytic wavelets and multiresolution analysis in different analytic function spaces. The extension of the wavelet theory to quaternion variable function spaces would be very beneficial in the solution of many problems in physics. A first step in this direction is to give the quaternionic analogue of the Blaschke group. In this paper we introduce the quaternionic Blaschke group and we study the properties of this group and its subgroups.

A Cohomological Property of π-invariant Elements

2013 ◽  
Vol 56 (3) ◽  
pp. 534-543
Author(s):  
M. Filali ◽  
M. Sangani Monfared
Keyword(s):  
Hilbert Space ◽  
Banach Algebra ◽  
Orthonormal Basis ◽  
Hochschild Cohomology ◽  
Separable Hilbert Space ◽  
Continuous Representation ◽  
Cohomology Groups ◽  
Cohomological Property ◽  
Coordinate Functions ◽  
Special Case

Abstract.Let A be a Banach algebra and let be a continuous representation of A on a separable Hilbert space H with dim H = m. Let πi j be the coordinate functions of π with respect to an orthonormal basis and suppose that for each and . Under these conditions, we call an element left π-invariant if In this paper we prove a link between the existence of left π-invariant elements and the vanishing of certain Hochschild cohomology groups of A. Our results extend an earlier result by Lau on F-algebras and recent results of Kaniuth, Lau, Pym, and and the second author in the special case where π : A → C is a non-zero character on A.

scholarly journals Besselian g-frames and near g-Riesz bases

2011 ◽  
Vol 5 (2) ◽  
pp. 259-270 ◽  
Author(s):  
M.R. Abdollahpour ◽  
A. Najati
Keyword(s):  
Hilbert Space ◽  
Riesz Basis ◽  
Orthonormal Basis ◽  
Inner Product ◽  
Riesz Bases ◽  
Finite Dimensional

In this paper we introduce and study near g-Riesz basis, Besselian g-frames and unconditional g-frames. We show that a near g-Riesz basis is a Besselian g-frame and we conclude that under some conditions the kernel of associated synthesis operator for a near g-Riesz basis is finite dimensional. Finally, we show that a g-frame is a g-Riesz basis for a Hilbert space H if and only if there is an equivalent inner product on H, with respect to which it becomes an g-orthonormal basis for H.

scholarly journals Hamiltonians Generated by Parseval Frames

2020 ◽  
Vol 171 (1) ◽  
Author(s):  
F. Bagarello ◽  
S. Kużel
Keyword(s):  
Hilbert Space ◽  
Riesz Basis ◽  
Physical System ◽  
Orthonormal Basis ◽  
Orthonormal Bases ◽  
Parseval Frames ◽  
Orthogonal Projectors ◽  
Physical Reasons ◽  
Math Phys ◽  
Infinite Linear

AbstractIt is known that self-adjoint Hamiltonians with purely discrete eigenvalues can be written as (infinite) linear combination of mutually orthogonal projectors with eigenvalues as coefficients of the expansion. The projectors are defined by the eigenvectors of the Hamiltonians. In some recent papers, this expansion has been extended to the case in which these eigenvectors form a Riesz basis or, more recently, a ${\mathcal{D}}$ D -quasi basis (Bagarello and Bellomonte in J. Phys. A 50:145203, 2017, Bagarello et al. in J. Math. Phys. 59:033506, 2018), rather than an orthonormal basis. Here we discuss what can be done when these sets are replaced by Parseval frames. This interest is motivated by physical reasons, and in particular by the fact that the mathematical Hilbert space where the physical system is originally defined, contains sometimes also states which cannot really be occupied by the physical system itself. In particular, we show what changes in the spectrum of the observables, when going from orthonormal bases to Parseval frames. In this perspective we propose the notion of $E$ E -connection for observables. Several examples are discussed.

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