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Communications on Pure & Applied Analysis

2007 , Volume 6 , Issue 3

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Preface
Tao Qiang and Yuesheng Xu
2007, 6(3): i-iii doi: 10.3934/cpaa.2007.6.3i +[Abstract](157) +[PDF](49.5KB)
Abstract:
This special issue contains a selection of 19 papers from two separate sources. About half of the total is from submissions to the international conference on Wavelet Analysis and Applications, 2005, University of Macau. The conference had 170 submissions from scholars and engineers from 22 different countries and areas, including Australia, Belgium, Brazil, China, Ethiopia, France, Germany, India, Iran, Hong Kong, Japan, Korea, Macao, Malaysia, Maxico, Portugal, Russia, Taiwan, Thailand, Tunisia, United Kingdom and United States. Papers selected for this issue are of top quality among the conference submissions and all contain substantial new results. The second source of this issue is from invited contributions from world-wide experts in the relevant areas. All the papers that appear in this volume are strictly refereed. We sincerely thank the referees for their extremely valuable assistance in creating this volume.

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A theoretical framework for wavelet analysis in a Hermitean Clifford setting
Fred Brackx, Hennie De Schepper and Frank Sommen
2007, 6(3): 549-567 doi: 10.3934/cpaa.2007.6.549 +[Abstract](211) +[PDF](279.7KB)
Abstract:
Hermitean Clifford analysis focusses on monogenic functions taking values in a complex Clifford algebra or in a complex spinor space. Here monogenicity is expressed by means of two complex mutually adjoint Dirac operators, which are invariant under the action of a representation of the unitary group. In this paper we have further developed the Hermitean theory by introducing so-called zonal functions and by studying plane wave null solutions of the Hermitean Dirac operators. Moreover we have defined new Hermite polynomials in this Hermitean setting and expressed them in terms of the former Clifford-Hermite polynomials and of the one-dimensional Laguerre polynomials. These Hermitean Hermite polynomials are good candidates for mother wavelets in a Hermitean continuous wavelet transformation theory yet to be developed.
Limits of radial basis function interpolants
Martin D. Buhmann and Slawomir Dinew
2007, 6(3): 569-585 doi: 10.3934/cpaa.2007.6.569 +[Abstract](257) +[PDF](217.9KB)
Abstract:
We solve some open problems posed by Fornberg et al. in [6], [9] and [12], related to radial basis functions with parameters. They concern the limits of interpolants using these radial basis functions when the aforementioned parameters tend to zero--which makes them "increasingly flat" in a term coined by Fornberg. These aspects of radial basis function interpolation are useful because they concern the numerical problems with ill-conditioned matrices for small parameters and how to solve the interpolation problems efficiently in the face of this ill-conditioning. Finally, there are some interesting links between radial basis function interpolation and polynomial interpolation coming out of this research. While answering several such conjectures, we also develop a number of new techniques--some of them with number-theoretic arguments--for attacking similar problems.
On the numerical evaluation of fractional Sobolev norms
Carsten Burstedde
2007, 6(3): 587-605 doi: 10.3934/cpaa.2007.6.587 +[Abstract](222) +[PDF](268.3KB)
Abstract:
In several important and active fields of modern applied mathematics, such as the numerical solution of PDE-constrained control problems or various applications in image processing and data fitting, the evaluation of (integer and real) Sobolev norms constitutes a crucial ingredient. Different approaches exist for varying ranges of smoothness indices and with varying properties concerning exactness, equivalence and the computing time for the numerical evaluation. These can usually be expressed in terms of discrete Riesz operators.
We propose a collection of criteria which allow to compare different constructions. Then we develop a unified approach which is valid for non-negative real smoothness indices for standard finite elements, and for positive and negative real smoothness for biorthogonal wavelet bases. This construction delivers a wider range of exactness than the currently known constructions and is computable in linear time.
Spectral Galerkin method for stochastic wave equations driven by space-time white noise
Yanzhao Cao and Li Yin
2007, 6(3): 607-617 doi: 10.3934/cpaa.2007.6.607 +[Abstract](244) +[PDF](150.3KB)
Abstract:
This paper is concerned with numerical approximations of stochastic wave equations driven by additive space-time white noise in one dimensional space. Convergence analysis and error estimates are presented for the numerical solutions based on the spectral Galerkin method with discretization in space variable. We obtain an estimate for the convergence rate. Comparing with the result of the finite difference approximation of Quer-Sardanyons and Sanz-Solé, the spectral Galerkin method enjoys higher convergence rate. Our error estimate is comparable to the error estimate of another finite difference scheme recently constructed by Walsh. However, our analysis is much simpler and our algorithm is easier to implemented.
Continuous wavelet transform and wavelet frames on the sphere using Clifford analysis
P. Cerejeiras, M. Ferreira, U. Kähler and F. Sommen
2007, 6(3): 619-641 doi: 10.3934/cpaa.2007.6.619 +[Abstract](190) +[PDF](455.9KB)
Abstract:
In this paper we construct a continuous wavelet transform (CWT) on the sphere $S^{n-1}$ based on the conformal group of the sphere, the Lorentz group Spin$(1,n)$. For this purpose, we present a short survey on the existing techniques of continuous wavelet transform and of conformal transformations on the unit sphere. We decompose the conformal group into the maximal compact subgroup of rotations Spin$(n)$ and the set of Möbius transformations of the form $\varphi_a(x) = (x-a)(1+ax)^{-1}$, where $a \in B^n$ and $B^n$ denotes the unit ball in $\mathbb R^n$. Based on a study of the influence of the parameter $a$ arising in the definition of dilations/contractions on the sphere we define a class of local conformal dilation operators and consequently a family of continuous wavelet transforms for the Hilbert space of square integrable functions on the sphere $L_2(S^{n-1})$ and the Hardy space $H^2$. In the end we construct Banach frames for our wavelets and prove Jackson-type theorems for the best $n$-point approximation.
Fast numerical collocation solutions of integral equations
Zhongying Chen, Bin Wu and Yuesheng Xu
2007, 6(3): 643-666 doi: 10.3934/cpaa.2007.6.643 +[Abstract](323) +[PDF](641.2KB)
Abstract:
We present a numerical implementation of a fast multiscale collocation method for solving Fredholm integral equations of the second kind with weakly singular kernels. The general setting of such a collocation method was recently developed by Chen, Micchelli and Xu. Following the general setting, in this paper we consider three important problems for the practical use of such collocation methods. The first problem regards the construction of concrete multiscale piecewise linear, quadratic and cubic polynomial functions and the corresponding multiscale collocation functionals. The second problem that we address is the practical truncation of the coefficient matrix. We propose a block truncation strategy which allows us to compress the matrix without computing the distances between the supports of a basis function and a collocation functional. The last problem is the fast numerical solution of the large discrete linear system resulting from the compression. We make use of the multiscale structure and the sparseness of the coefficient matrix in developing fast solver for the linear system. Numerical examples are presented to demonstrate the accuracy and computational speed of the method.
Full rank filters and polynomial reproduction
Mariantonia Cotronei and Tomas Sauer
2007, 6(3): 667-687 doi: 10.3934/cpaa.2007.6.667 +[Abstract](240) +[PDF](268.7KB)
Abstract:
We study polynomial (re)production properties of full rank stationary subdivision schemes and develop characterizations that are almost identical to the scalar case. Moreover, we give a version of the Strang--Fix conditions for the integer translates of a matrix valued function whose columns are linearly independent.
Analysis of optimal bivariate symmetric refinable Hermite interpolants
Bin Han and Qun Mo
2007, 6(3): 689-718 doi: 10.3934/cpaa.2007.6.689 +[Abstract](241) +[PDF](371.1KB)
Abstract:
Multivariate refinable Hermite interpolants with high smoothness and small support are of interest in CAGD and numerical algorithms. In this article, we are particularly interested in analyzing some univariate and bivariate symmetric refinable Hermite interpolants, which have some desirable properties such as short support, optimal smoothness and spline property. We shall study the projection method for multivariate refinable function vectors and discuss some properties of multivariate spline refinable function vectors. Here a compactly supported multivariate spline function on $\mathbb R^s$ just means a function of piecewise polynomials supporting on a finite number of polygonal partition subdomains of $\mathbb R^s$. We shall discuss spline refinable function vectors by investigating the structure of the eigenvalues and eigenvectors of the transition operator. To illustrate the results in this paper, we shall analyze the optimal smoothness and spline properties of some univariate and bivariate refinable Hermite interpolants. For the regular triangular mesh, we obtain a bivariate $C^2$ symmetric dyadic refinable Hermite interpolant of order $2$ whose mask is supported inside $[-1,1]^2$.
Error analysis of the p-version discontinuous Galerkin method for heat transfer in built-up structures
Kim S. Bey, Peter Z. Daffer, Hideaki Kaneko and Puntip Toghaw
2007, 6(3): 719-740 doi: 10.3934/cpaa.2007.6.719 +[Abstract](249) +[PDF](408.4KB)
Abstract:
In this paper, we provide an error analysis for the $p$-version of the discontinuous Galerkin finite element method for a class of heat transfer problems in built-up structures. Also, a general form of the matrix associated with the discretization of time variable using the $p$-finite element basis functions is established. Many interesting properties of this matrix are obtained. Numerical examples are provided in the last section.
Reducing subspace frame multiresolution analysis and frame wavelets
Qiao-Fang Lian and Yun-Zhang Li
2007, 6(3): 741-756 doi: 10.3934/cpaa.2007.6.741 +[Abstract](271) +[PDF](212.8KB)
Abstract:
Multiresolution analysis (MRA) and frame multiresolution analysis (FMRA) in $L^2(\mathbb R)$ play a significant role in the construction of wavelets and frame wavelets for $L^2(\mathbb R)$. In this paper, the notions of MRA and FMRA in a reducing subspace of $L^2(\mathbb R)$ are introduced, from which the construction of wavelets and frame wavelets for this subspace is obtained. Many examples are also provided to illustrate the general theory. In particular, most of them are about the space with its frequency lying in $[0,\infty)$, which is closely related to various Paley-Wiener Theorems.
Localizations and parallelizations for two-scale finite element discretizations
Fang Liu and Aihui Zhou
2007, 6(3): 757-773 doi: 10.3934/cpaa.2007.6.757 +[Abstract](216) +[PDF](238.3KB)
Abstract:
Some local and parallel algorithms for two-scale finite element discretizations are proposed and analyzed in this paper for elliptic boundary value problems. These algorithms are motivated by the observation that, for a solution to some elliptic boundary value problems, low frequency components can be approximated well by a relatively coarse grid and high frequency components can be computed on partially fine grids by some local procedure. A theoretical tool for analyzing these algorithms is some recent local error estimates for finite element approximations.
Refinable functions on the Heisenberg group
Heping Liu and Yu Liu
2007, 6(3): 775-787 doi: 10.3934/cpaa.2007.6.775 +[Abstract](200) +[PDF](172.6KB)
Abstract:
The orthonormal wavelets associated with a multiresolution analysis are mainly determined by the corresponding refinable function. In this paper, we study the continuity of refinable functions on the Heisenberg group. The characterization of Lipschitz continuous refinable functions is given in terms of the uniform joint spectral radius. We also give an investigation of the refinable functions in the generalized Lipschitz spaces.
Interpolating filters with prescribed zeros and their refinable functions
C. A. Micchelli and Q. Sun
2007, 6(3): 789-808 doi: 10.3934/cpaa.2007.6.789 +[Abstract](214) +[PDF](253.3KB)
Abstract:
In this paper, we study the minimally supported interpolating filters with prescribed zeros and their corresponding refinable functions.
Refinable functions with general dilation and a stable test for generalized Routh-Hurwitz conditions
J. M. Peña
2007, 6(3): 809-818 doi: 10.3934/cpaa.2007.6.809 +[Abstract](221) +[PDF](156.8KB)
Abstract:
Generalized Routh-Hurwitz conditions consist of the positivity of $n$ determinants associated to a polynomial of degree $n$. They can be used in order to guarantee that a refinable function with dilation $M$ is a ripplet, that is, the collocation matrices of its shifts are totally positive. Given a polynomial of degree $n$, a test of $\mathcal O(n^2)$ elementary operations and growth factor 1 is presented in order to check the generalized Routh-Hurwitz conditions. The case corresponding to $M=3$ is described in detail.
Localization operator and digital communication capacity of channel
Lizhong Peng, Shujun Dang and Bojin Zhuang
2007, 6(3): 819-827 doi: 10.3934/cpaa.2007.6.819 +[Abstract](249) +[PDF](132.2KB)
Abstract:
In mathematical language, the communication model of some digital information channel can be described as the self-adjoint operator $Q_T P_\Omega Q_T$ on $L^2(R)$, where $T$ and $\Omega$ are constants. $T$ is called signal peroid and $\Omega$ is called channel's bandwidth. By computing the eigenvalues and the corresponding eigenfunctions of the compact self-adjoint operator $Q_T P_\Omega Q_T$, the conclusion of Landau, Pollak and Slepian shows that about $2\Omega T$ bits data can be transmitted by this channel within time $T$ when $\Omega T$ is sufficiently large. Considering the realistic communication model, this paper points out that in one signal period $T$, at most $2r\Omega T$ can be transmitted with $r$ ($r\in (0,1)$) dependent on some given threshold $\eta $.
Parity symmetry in multi-dimensional signals
Gerald Sommer and Di Zang
2007, 6(3): 829-852 doi: 10.3934/cpaa.2007.6.829 +[Abstract](257) +[PDF](977.7KB)
Abstract:
Parity symmetry is an important local feature for qualitative signal analysis. It is strongly related to the local phase of the signal. In image processing parity symmetry is a cue for the line-like or edge-like quality of a local image structure. The analytic signal is a well-known representation for 1D signals, which enables the extraction of local spectral representations as amplitude and phase. Its representation domain is that of the complex numbers. We will give an overview how the analytic signal can be generalized to the monogenic signal in the $n$D case within a Clifford valued domain. The approach is based on the Riesz transform as a generalization of the Hilbert transform with respect to the embedding dimension of the structure. So far we realized the extension to 2D and 3D signals. We learned to take advantage of interesting effects of the proposed generalization as the simultaneous estimation of the local amplitude, phase and orientation, and of image analysis in the monogenic scale-space.
The complete orthogonal V-system and its applications
Hui Ma, Dongxu Qi, Ruixia Song and Tianjun Wang
2007, 6(3): 853-871 doi: 10.3934/cpaa.2007.6.853 +[Abstract](205) +[PDF](1322.2KB)
Abstract:
Based on the function generator on [0,1], a class of complete orthogonal function system called as the V-system is studied in this paper. The V-system is composed by piecewise polynomials, and is capable of exactly describing the geometric information expressed by the popularly and widely used polynomial spline curves and surfaces. The V-system has all of the beautiful properties of the U-system: continuity, discontinuity, orthogonal completeness and reproducibility. In addition, the V-system also has the concise structure, compactly local support and multi-resolution capability. The V-system is the generalization of the well-known Haar function system, and is also a new class of practical and flexible wavelet bases. By utilizing the concepts of the energy and the descriptor of the V-system, we study the degree of similarity of geometric models which can be used in image analysis and processing.
Wavelet approach to numerical differentiation of noisy functions
Jianzhong Wang
2007, 6(3): 873-897 doi: 10.3934/cpaa.2007.6.873 +[Abstract](200) +[PDF](225.6KB)
Abstract:
We apply wavelet transform in the study of numerical differentiation for the functions which are infected by noise. Because of the presence of noise, the observed noisy function is not differentiable. In order to estimate the derivatives of the target function from its observation, a pretreatment of the observation is necessary. The paper introduces differential approximation wavelets (DA-wavelets) so that the DA-wavelet transforms of the observed function approximate the derivatives of the target function. The paper also shows that the derivatives of compactly supported splines lead to a certain type of DA-wavelet transforms, which are difference formulas for computing derivatives. The relation between difference formulas and splines enables us to construct various difference formulas via splines and to estimate the computing errors of difference formulas in the spline framework.
Edge detection by using rotational wavelets
Liming Zhang, Tao Qian and Qingye Zeng
2007, 6(3): 899-915 doi: 10.3934/cpaa.2007.6.899 +[Abstract](241) +[PDF](435.5KB)
Abstract:
Based on a mathematical model involving Radon measure explicit computations on convolution integrals defining continuous (integral) wavelet transformations are carried out. The study shows that the truncated Morlet wavelet significantly depends on a rotation parameter and thus lay a foundation of edge detection in pattern recognition and image processing using rotational (directional) wavelets. Experiments and algorithms are developed based on the theory. The theory is further generalized to the $n$-dimensional cases and to a large class of rotational wavelets.

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