THE IMAGE SPACE OF ONE TYPE OF CONTINUOUS WAVELET TRANSFORM AND ITS PROPERTY

2012 ◽  
Vol 10 (03) ◽  
pp. 1250022 ◽  
Author(s):  
CAIXIA DENG ◽  
ZUOXIAN FU ◽  
SHUAI LI
Keyword(s):  
Hilbert Space ◽  
Wavelet Transform ◽  
Continuous Wavelet Transform ◽  
Fundamental Theorem ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Image Space ◽  
Continuous Wavelet ◽  
Admissible Wavelet ◽  
Linear Transform

In this paper, we show that the space of continuous wavelet transform is a reproducing kernel Hilbert space based on the fundamental theorem of linear transform. An admissible wavelet is got by convolution computation which is made into continuous wavelet transform. By the theory of reproducing kernel we can discuss correlative properties of image space of wavelet transform, which provide theoretic frame for us to study image space of the general wavelet transform.

ON THE CONTINUOUS WAVELET TRANSFORM ON HOMOGENEOUS SPACES

2012 ◽  
Vol 10 (04) ◽  
pp. 1250038 ◽  
Author(s):  
F. ESMAEELZADEH ◽  
R. A. KAMYABI GOL ◽  
R. RAISI TOUSI
Keyword(s):  
Hilbert Space ◽  
Wavelet Transform ◽  
Continuous Wavelet Transform ◽  
Reproducing Kernel ◽  
Homogeneous Spaces ◽  
Reproducing Kernel Hilbert Space ◽  
Compact Subgroup ◽  
Continuous Wavelet ◽  
Square Integrable Representation ◽  
Necessary And Sufficient

Let G be a locally compact group with a compact subgroup H. We define a square integrable representation of a homogeneous space G/H on a Hilbert space [Formula: see text]. The reconstruction formula for G/H is established and as a result it is concluded that the set of admissible vectors is path connected. The continuous wavelet transform on G/H is defined and it is shown that the range of the continuous wavelet transform is a reproducing kernel Hilbert space. Moreover, we obtain a necessary and sufficient condition for the continuous wavelet transform to be onto.

CHARACTERIZATION OF IMAGE SPACE OF A WAVELET TRANSFORM

2006 ◽  
Vol 04 (03) ◽  
pp. 547-557 ◽  
Author(s):  
CAIXIA DENG ◽  
YULING QU ◽  
LIJUAN GU
Keyword(s):  
Wavelet Transform ◽  
Truncation Error ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Sampling Theorem ◽  
Image Space ◽  
Kernel Space ◽  
Reproducing Kernel Space ◽  
Scope Of Application

In this paper, Journe wavelet function is introduced as a wavelet generating function. The expression of reproducing kernel function for the image space of this wavelet transform is obtained based on the fact that the image space of the wavelet transform is a reproducing kernel Hilbert space. Then the isometric identity of Journe wavelet transform is obtained. The connections between the image space of the wavelet transform and the image space of the known reproducing kernel space are established by the theories of reproducing kernel. The properties and the structures of the image space of the wavelet transform can be characterized by the properties and the structures of the image space of the known reproducing kernel space. Using the ideas of reproducing kernel, we consider there are relations between the wavelet transform and the sampling theorem. Meanwhile, the approximations in sampling theorems is shown and the truncation error is given. This provides a theoretical basis for us to study the image space of the general wavelet transform and broadens the scope of application of theories of the reproducing kernel space.

scholarly journals Reproducing kernel Hilbert spaces of CR functions for the Euclidean motion group

2015 ◽  
Vol 13 (03) ◽  
pp. 331-346 ◽  
Author(s):  
D. Barbieri ◽  
G. Citti
Keyword(s):  
Hilbert Space ◽  
Wavelet Transform ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Bargmann Transform ◽  
Euclidean Motion Group ◽  
Cr Functions ◽  
Kernel Hilbert Spaces ◽  
Euclidean Motion ◽  
Hilbert Subspace

We study the geometric structure of the reproducing kernel Hilbert space associated to the continuous wavelet transform generated by the irreducible representations of the group of Euclidean motions of the plane SE(2). A natural Hilbert norm for functions on the group is constructed that makes the wavelet transform an isometry, but since the considered representations are not square integrable, the resulting Hilbert space will not coincide with L2( SE (2)). The reproducing kernel Hilbert subspace generated by the wavelet transform, for the case of a minimal uncertainty mother wavelet, can be characterized in terms of the complex regularity defined by the natural CR structure of the group. Relations with the Bargmann transform are presented.

scholarly journals Lebesgue Decomposition of Non-Commutative Measures

2020 ◽  
Author(s):  
Michael T Jury ◽  
Robert T W Martin
Keyword(s):  
Hilbert Space ◽  
Reproducing Kernel ◽  
Fock Space ◽  
Reproducing Kernel Hilbert Space ◽  
Semigroup Algebra ◽  
Space Theory ◽  
Lebesgue Decomposition ◽  
Sesquilinear Forms ◽  
Positive Linear Functionals ◽  
Algebra Theory

Abstract We extend the Lebesgue decomposition of positive measures with respect to Lebesgue measure on the complex unit circle to the non-commutative (NC) multi-variable setting of (positive) NC measures. These are positive linear functionals on a certain self-adjoint subspace of the Cuntz–Toeplitz $C^{\ast }-$algebra, the $C^{\ast }-$algebra of the left creation operators on the full Fock space. This theory is fundamentally connected to the representation theory of the Cuntz and Cuntz–Toeplitz $C^{\ast }-$algebras; any *−representation of the Cuntz–Toeplitz $C^{\ast }-$algebra is obtained (up to unitary equivalence), by applying a Gelfand–Naimark–Segal construction to a positive NC measure. Our approach combines the theory of Lebesgue decomposition of sesquilinear forms in Hilbert space, Lebesgue decomposition of row isometries, free semigroup algebra theory, NC reproducing kernel Hilbert space theory, and NC Hardy space theory.

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