scholarly journals The Exponential Sampling Theorem of Signal Analysis and the Reproducing Kernel Formula in the Mellin Transform Setting

2014 ◽  
Vol 13 (1) ◽  
pp. 35-66 ◽  
Author(s):  
Carlo Bardaro ◽  
Paul Leo Butzer ◽  
Ilaria Mantellini
Keyword(s):  
Signal Analysis ◽  
Mellin Transform ◽  
Reproducing Kernel ◽  
Sampling Theorem ◽  
Reproducing Kernel Formula

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 kernels of Sobolev spaces on ℝd and applications to embedding constants and tractability

2018 ◽  
Vol 16 (05) ◽  
pp. 693-715 ◽  
Author(s):  
Erich Novak ◽  
Mario Ullrich ◽  
Henryk Woźniakowski ◽  
Shun Zhang
Keyword(s):  
Sobolev Space ◽  
Sobolev Spaces ◽  
Reproducing Kernel ◽  
Absolute Error ◽  
Reproducing Kernels ◽  
Worst Case ◽  
Reproducing Kernel Formula ◽  
Positive Integers ◽  
Integration Errors ◽  
Exponentially Small

The standard Sobolev space [Formula: see text], with arbitrary positive integers [Formula: see text] and [Formula: see text] for which [Formula: see text], has the reproducing kernel [Formula: see text] for all [Formula: see text], where [Formula: see text] are components of [Formula: see text]-variate [Formula: see text], and [Formula: see text] with non-negative integers [Formula: see text]. We obtain a more explicit form for the reproducing kernel [Formula: see text] and find a closed form for the kernel [Formula: see text]. Knowing the form of [Formula: see text], we present applications on the best embedding constants between the Sobolev space [Formula: see text] and [Formula: see text], and on strong polynomial tractability of integration with an arbitrary probability density. We prove that the best embedding constants are exponentially small in [Formula: see text], whereas worst case integration errors of algorithms using [Formula: see text] function values are also exponentially small in [Formula: see text] and decay at least like [Formula: see text]. This yields strong polynomial tractability in the worst case setting for the absolute error criterion.

Discrete Mellin transform for signal analysis

2002 ◽  
Author(s):  
J. Bertrand ◽  
P. Bertrand ◽  
J.P. Ovarlez
Keyword(s):  
Signal Analysis ◽  
Mellin Transform

scholarly journals Second-Order Linear Differential Equations with Solutions in Analytic Function Spaces

2019 ◽  
Vol 2019 ◽  
pp. 1-9
Author(s):  
Jianren Long ◽  
Yu Sun ◽  
Shimei Zhang ◽  
Guangming Hu
Keyword(s):  
Analytic Function ◽  
Reproducing Kernel ◽  
Bloch Space ◽  
Sufficient Conditions ◽  
Representation Formula ◽  
Second Order ◽  
Order Linear ◽  
Linear Differential ◽  
The One ◽  
Reproducing Kernel Formula

This research is concerned with second-order linear differential equation f′′+A(z)f=0, where A(z) is an analytic function in the unit disc. On the one hand, some sufficient conditions for the solutions to be in α-Bloch (little α-Bloch) space are found by using exponential type weighted Bergman reproducing kernel formula. On the other hand, we find also some sufficient conditions for the solutions to be in analytic Morrey (little analytic Morrey) space by using the representation formula.

Sampling analysis in the complex reproducing kernel Hilbert space

2014 ◽  
Vol 26 (1) ◽  
pp. 109-120 ◽  
Author(s):  
BING-ZHAO LI ◽  
QING-HUA JI
Keyword(s):  
Hilbert Space ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Sampling Theorem ◽  
Optimal Approximation ◽  
Sampling Points ◽  
Limited Function ◽  
Band Limited ◽  
Band Limited Function ◽  
Necessary And Sufficient

We consider and analyse sampling theories in the reproducing kernel Hilbert space (RKHS) in this paper. The reconstruction of a function in an RKHS from a given set of sampling points and the reproducing kernel of the RKHS is discussed. Firstly, we analyse and give the optimal approximation of any function belonging to the RKHS in detail. Then, a necessary and sufficient condition to perfectly reconstruct the function in the corresponding RKHS of complex-valued functions is investigated. Based on the derived results, another proof of the sampling theorem in the linear canonical transform (LCT) domain is given. Finally, the optimal approximation of any band-limited function in the LCT domain from infinite sampling points is also analysed and discussed.

scholarly journals Aspects of prediction

2014 ◽  
Vol 51 (A) ◽  
pp. 189-201
Author(s):  
N. H. Bingham ◽  
Badr Missaoui
Keyword(s):  
Hilbert Spaces ◽  
Continuous Time ◽  
Reproducing Kernel ◽  
Sampling Theorem ◽  
Reproducing Kernel Hilbert Spaces ◽  
Stationary Processes ◽  
Prediction Theory ◽  
Volatility Clustering ◽  
Infinite Dimensional ◽  
Kernel Hilbert Spaces

We survey some aspects of the classical prediction theory for stationary processes, in discrete time in Section 1, turning in Section 2 to continuous time, with particular reference to reproducing-kernel Hilbert spaces and the sampling theorem. We discuss the discrete-continuous theories of ARMA-CARMA, GARCH-COGARCH, and OPUC-COPUC in Section 3. We compare the various models treated in Section 4 by how well they model volatility, in particular volatility clustering. We discuss the infinite-dimensional case in Section 5, and turn briefly to applications in Section 6.

scholarly journals Inhomogeneous shearlet coorbit spaces

2018 ◽  
Vol 16 (04) ◽  
pp. 1850026
Author(s):  
Fabian Feise ◽  
Lukas Sawatzki
Keyword(s):  
Banach Spaces ◽  
Reproducing Kernel ◽  
Lebesgue Spaces ◽  
Shearlet Transform ◽  
Weighted Lebesgue Spaces ◽  
Coorbit Spaces ◽  
Reproducing Kernel Formula

In this paper, we establish inhomogeneous coorbit spaces related to the continuous shearlet transform and the weighted Lebesgue spaces [Formula: see text] for certain weights [Formula: see text]. We present an inhomogeneous shearlet frame for [Formula: see text] which gives rise to a reproducing kernel [Formula: see text] that is not contained in the space [Formula: see text]. We show that the inhomogeneous shearlet coorbit spaces are Banach spaces by introducing a generalization of the approach of Fornasier, Rauhut and Ullrich.

Sampling of bandlimited signals in the offset linear canonical transform domain based on reproducing kernel Hilbert space

2019 ◽  
Vol 18 (02) ◽  
pp. 1950054 ◽  
Author(s):  
Shuiqing Xu ◽  
Zhiwei Chen ◽  
Yi Chai ◽  
Yigang He ◽  
Xiang Li
Keyword(s):  
Hilbert Space ◽  
Reproducing Kernel ◽  
Reproducing Kernel Hilbert Space ◽  
Sampling Theorem ◽  
Orthogonal Basis ◽  
Transform Domain ◽  
Linear Canonical Transform ◽  
Uniform Sampling ◽  
Bandlimited Signals ◽  
Offset Linear Canonical Transform

The offset linear canonical transform (OLCT) has proven to be a novel and effective method in signal processing and optics. Many important properties and results of the OLCT have been well studied and published. In this work, the sampling theorem of the OLCT bandlimited signals based on reproducing kernel Hilbert space has been proposed. First, we show that the bandlimited signals in the OLCT domain form a reproducing kernel Hilbert space. Then, an orthogonal basis for the OLCT bandlimited signals has been obtained based on the reproducing kernel Hilbert space. By using the orthogonal basis, the uniform sampling theory for bandlimited signals associated with the OLCT has been obtained. Furthermore, the nonuniform sampling of the OLCT bandlimited signals also has been attained. Finally, the simulations are provided to prove the usefulness and correctness of the derived results.

scholarly journals Aspects of prediction

2014 ◽  
Vol 51 (A) ◽  
pp. 189-201 ◽  
Author(s):  
N. H. Bingham ◽  
Badr Missaoui
Keyword(s):  
Hilbert Spaces ◽  
Continuous Time ◽  
Reproducing Kernel ◽  
Sampling Theorem ◽  
Reproducing Kernel Hilbert Spaces ◽  
Stationary Processes ◽  
Prediction Theory ◽  
Volatility Clustering ◽  
Infinite Dimensional ◽  
Kernel Hilbert Spaces

We survey some aspects of the classical prediction theory for stationary processes, in discrete time in Section 1, turning in Section 2 to continuous time, with particular reference to reproducing-kernel Hilbert spaces and the sampling theorem. We discuss the discrete-continuous theories of ARMA-CARMA, GARCH-COGARCH, and OPUC-COPUC in Section 3. We compare the various models treated in Section 4 by how well they model volatility, in particular volatility clustering. We discuss the infinite-dimensional case in Section 5, and turn briefly to applications in Section 6.

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