NEW WAVELET BASES FOR NON－HOMOGENEOUS SYMBOLIC SPACE OpS1,1^m AND RELATED KERNEL－DISTRIBUTION SPACES

This paper constructs several classes of new wavelet bases, which are unconditional bases for related operator spaces. Using these bases, the author analyzes non-homogeneous symbolic space OpSm1,1 and two related kernel-distribution spaces, and characterizes them in two wavelet coefficients spaces. Besides, some properties for singular integral operators are studied.     

BASES OF BIVARIATE SPLINE SPACES WITH CROSS-CUT GRID PARTITIONS

In this paper we determine the dimensions of certain bivariate spline spaces with cross-cut grid partitions and give explicit expressions of their bases functions. As an immediate consequence, the closures of these spaces over all grid partitions of the same type are known. The results in this paper can be applied to interpolation and approximation by bivariate spline functions.     

ON SPACES OF FRACTAL FUNCTIONS

In this paper we Ointroduce linear-spaces consisting of continuous functions whose graphs are the attactars of a special class of iterated function systems. We show that such spaces are finite dimensional and give the bases of these spaces in an implicit way. Given such a space, we discuss how to obtain a set of knots for whah the Lagrange interpolation problem by the space is uniquely solvable.     

A quantum analogue of generic bases for affine cluster algebras

We construct quantized versions of generic bases in quantum cluster algebras of finite and affine types.Under the specialization of q and coefficients to 1,these bases are generic bases of finite and affine cluster algebras.     

基于L~1度量分析的图象分解与重构算法

Mallat‘s decompositon and reconstruction algorithms are very important in the the field of wavelet theory and its applications to signal processing.Wavelet Anal-ysis,which is based on L^2(R) space,can eliminate redundancy of signals with the help of orthogonality and characterize the processing precision with the meansquare error.In the recent years,it is understood that the mean square measuredoes not match human visual sensitivity well.From the point of view,R.DeVore studied L^1 measure instead.Similarly,considering the principles of image com-pression,Yang introduced and dealt with orthogonality in L^1 space based on thebest approximation theory,and consequently established the corresponding decom-position and reconstruction algorithms for signals.In this paper,error analyses for the algorithms above are taken and the selection of the best parameters in the algorithms are discussed in detail.Finally,the algorithms are compared with the classical Haar and Daubechies‘‘s orthogonal wavelets based on the singal-to-noiseratio data computed.     

SPECTRAL APPROXIMATION ORDERS OF MULTIDIMENSIONAL NONSTATIONARY BIORTHOGONAL SEMI-MULTIRESOLUTION ANALYSIS IN SOBOLEV SPACE

Subdivision algorithm （Stationary or Non-stationary） is one of the most active and exciting research topics in wavelet analysis and applied mathematical theory. In multidimensional non-stationary situation, its limit functions are both compactly supported and infinitely differentiable. Also, these limit functions can serve as the scaling functions to generate the multidimensional non-stationary orthogonal or biorthogonal semi-multiresolution analysis （Semi-MRAs）. The spectral approximation property of multidimensional non-stationary biorthogonal Semi-MRAs is considered in this paper. Based on nonstationary subdivision scheme and its limit scaling functions, it is shown that the multidimensional nonstationary biorthogonal Semi-MRAs have spectral approximation order r in Sobolev space H^s（R^d）, for all r ≥ s ≥ 0.     

Coefficient Inequalities for p-Valent Functions

In the present paper, the authors introduce a new subclass of p-valent analytic functions with complex order defined on the open unit disk U={z:z∈C and |z|1} and obtain coefficient inequalities for the functions in these class. Application of these results for the functions defined by the convolution are also obtained.     

Binary image reconstruction based on prescribed numerical information

The problem of reconstruction of a binary image in the field of discrete tomography is a classic instance of seeking solution applying mathematical techniques. Here two such binary image reconstruction problems are considered given some numerical information on the image. Algorithms are developed for solving these problems and correctness of the algorithms are discussed.     

Tree wavelet approximations with applications

We construct a tree wavelet approximation by using a constructive greedy scheme (CGS). We define a function class which contains the functions whose piecewise polynomial approximations generated by the CGS have a prescribed global convergence rate and establish embedding properties of this class. We provide sufficient conditions on a tree index set and on bi-orthogonal wavelet bases which ensure optimal order of convergence for the wavelet approximations encoded on the tree index set using the bi-orthogonal wavelet bases. We then show that if we use the tree index set associated with the partition generated by the CGS to encode a wavelet approximation, it gives optimal order of convergence.     

ON THE INVERSE IMAGE SET OF RATIONAL FUNCTIONS

This article studies the inverse image of rational functions.Several theorems are obtained on the Julia set expressed by the inverse image,and a mistake is pointed out in H.Brolin' theorem incidentally.     

A characterization ofN-dimensional daubechies type tensor product wavelet

In this paper, we consider the problem of the existence of general non-separable variate orthonormal compactly supported wavelet basis when the symbol function has a special form. We prove that the general non-separable variate orthonormal wavelet basis doesn't exist if the symbol function possesses a certain form. This helps us to explicate the difficulty of constructing the non-separable variate wavelet basis and to hint how to construct non-separable variate wavelet basis. This research is supported by the National Natural Science Foundation of China (No.69982002) and the Opening Foundation of National Mobile Communications Research Laboratory in Southeast University.     

Band-limited Wavelets and Framelets in Low Dimensions

In this paper, we study the problem of constructing non-separable band-limited wavelet tight frames, Riesz wavelets and orthonormal wavelets in $\mathbb {R}^{2}$ and $\mathbb {R}^{3}$ . We first construct a class of non-separable band-limited refinable functions in low-dimensional Euclidean spaces by using univariate Meyer’s refinable functions along multiple directions defined by classical box-spline direction matrices. These non-separable band-limited definable functions are then used to construct non-separable band-limited wavelet tight frames via the unitary and oblique extension principles. However, these refinable functions cannot be used for constructing Riesz wavelets and orthonormal wavelets in low dimensions as they are not stable. Another construction scheme is then developed to construct stable refinable functions in low dimensions by using a special class of direction matrices. The resulting stable refinable functions allow us to construct a class of MRA-based non-separable band-limited Riesz wavelets and particularly band-limited orthonormal wavelets in low dimensions with small frequency support.     

Examples of refinable componentwise polynomials

This short note presents four examples of compactly supported symmetric refinable componentwise polynomial functions: (i) a componentwise constant interpolatory continuous refinable function and its derived symmetric tight wavelet frame; (ii) a componentwise constant continuous orthonormal and interpolatory refinable function and its associated symmetric orthonormal wavelet basis; (iii) a differentiable symmetric componentwise linear polynomial orthonormal refinable function; (iv) a symmetric refinable componentwise linear polynomial which is interpolatory and differentiable.     

基于小波神经网络的NLAR(p)过程的逼近

小波神经网络是近年来发展起来的一种逼近非线性函数的新型人工神经网络。特别是，正交尺度函数为某函数的小波神经网络更适合于函数逼近。本文在此基础上讨论了小波神经网络对非线性AR(p)过程的逼近。     

二元可分正交紧支集小波基的刻划

本文讨论了当特征函数具有某种特殊形式时,不可分二元紧支集正交小波基的存在性问题．结论为,当特征函数为所给的形式时,不可分二元紧文集正交小波基是不存在的．     

正交小波变换的向量空间算法

本文揭示了一个事实，小波不仅可构成Ｌ２空间中的正交基，小波分解与重构滤波还可产生Ｎ维空间中的正交基．在本文提出修改的小波变换算法之下，Ｎ点信号的小波变换等价于Ｎ维空间中的正交变换．用该算法进行信号或图象压缩，无需对信号或图象进行周期延拓，可严格地在Ｎ维空间中进行．     

Lattice Structure for Paraunitary Linear-phase Filter Banks with Accuracy

Multivariate filter banks with a polyphase matrix built by matrix factorization （lattice structure） were proposed to obtain orthonormal wavelet basis. On the basis of that, we propose a general method of constructing filter banks which ensure second and third accuracy of its corresponding scaling function. In the last part, examples with second and third accuracy are given.     

Stability theorems for Fourier frames and wavelet Riesz bases

In this paper we present two applications of a Stability Theorem of Hilbert frames to nonharmonic Fourier series and wavelet Riesz basis. The first result is an enhancement of the Paley-Wiener type constant for nonharmonic series given by Duffin and Schaefer in [6] and used recently in some applications (see [3]). In the case of an orthonormal basis, our estimate reduces to Kadec’ optimal 1/4 result. The second application proves that a phenomenon discovered by Daubechies and Tchamitchian [4] for the orthonormal Meyer wavelet basis (stability of the Riesz basis property under small changes of the translation parameter) actually holds for a large class of wavelet Riesz bases.     

Radial Multiresolution in Dimension Three

We present a construction of a wavelet-type orthonormal basis for the space of radial $L^2$-functions in {\bf R}$^3$ via the concept of a radial multiresolution analysis. The elements of the basis are obtained from a single radial wavelet by usual dilations and generalized translations. Hereby the generalized translation reveals the group convolution of radial functions in {\bf R}$^3$. We provide a simple way to construct a radial scaling function and a radial wavelet from an even classical scaling function on {\bf R}. Furthermore, decomposition and reconstruction algorithms are formulated.     

Singularity spectrum of multifractal functions involving oscillating singularities

We give general mathematical results concerning oscillating singularities and we study examples of functions composed only of oscillating singularities. These functions are defined by explicit coefficients on an orthonormal wavelet basis. We compute their Hölder regularity and oscillation at every point and we deduce their spectrum of oscillating singularities.