Super‐wavelets on local fields of positive characteristic

The concept of super‐wavelet was introduced by Balan, and Han and Larson over the field of real numbers which has many applications not only in engineering branches but also in different areas of mathematics. To develop this notion on local fields having positive characteristic we obtain characterizations of super‐wavelets of finite length as well as Parseval frame multiwavelet sets of finite order in this setup. Using the group theoretical approach based on coset representatives, further we establish Shannon type multiwavelet in this perspective while providing examples of Parseval frame (multi)wavelets and (Parseval frame) super‐wavelets. In addition, we obtain necessary conditions for decomposable and extendable Parseval frame wavelets associated to Parseval frame super‐wavelets.     

Unconditional basis recursively generated in L p on compact sets

We construct an unconditional basis in the Banach space L ^p(Ω) for p 1 by using the refinement equation and the basic operation of translation and scale, where Ω is a compact subset in ℝⁿ. We also give an algorithm of how to construct an unconditional basis in L ^p(Ω p). At the end of this paper, we give the characterization of the functions in L ^p(Ω p) by using the wavelet coefficients.     

二进小波的构造方法研究

提出两种二进小波的构造方法.首先,将Mallat构造的B-样条二进小波推广得到一种构造B-样条二进小波的新方法;其次,基于二进提升方案提出构造二进小波的另一种新方法—–构造定理,并通过调整定理中提升参数的形式、以新的B-样条二进小波作为初始二进小波,具体构造了具有有限长单位脉冲响应、高阶消失矩、线性相位的提升二进小波,这些提升二进小波不能由Sweldens提升方案得到.     

一类非线性微分方程组的有理化Haar小波解法

利用有理化Haar小波性质和方法,建立了一类非线性微分方程组在任意区间[a,b）的求解算法．基于该算法,运用计算机代数系统Maple,给出了求解非线性微分方程组的程序．并运用此程序给出了一类微分方程组的计算实例,从数值模拟来看可以达到较高的精度,并对方程组的动力学行为给出较好的描述．     

Local analysis,cardinality, and split trick of quasi-biorthogonal frame wavelets

The notion of quasi-biorthogonal frame wavelets is a generalization of the notion of orthogonal wavelets. A quasi-biorthogonal frame wavelet with the cardinality r consists of r pairs of functions. In this paper we first analyze the local property of the quasi-biorthogonal frame wavelet and show that its each pair of functions generates reconstruction formulas of the corresponding subspaces. Next we show that the lower bound of its cardinalities depends on a pair of dual frame multiresolution analyses deriving it. Finally, we present a split trick and show that any quasi-biorthogonal frame wavelet can be split into a new quasi-biorthogonal frame wavelet with an arbitrarily large cardinality. For generality, we work in the setting of matrix dilations.     

Dimensional upper bounds for admissible subgroups for the metaplectic representation

We prove dimensional upper bounds for admissible Lie subgroups H of G = ?^d ? Sp (d, ?), d ≥ 2. The notion of admissibility captures natural geometric phenomena of the phase space and it is a sufficient condition for a subgroup to be reproducing. It is expressed in terms of absolutely convergent integrals of Wigner distributions, translated by the affine action of the subgroup. We show that dim H ≤ d² + 2d, whereas if H ? Sp (d, R), then dim H ≤ d² + 1. Both bounds are shown to be optimal (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Orthogonal systems of spline wavelets as unconditional bases in Sobolev spaces

We exhibit the necessary range for which functions in the Sobolev spaces $L_p^s$ can be represented as an unconditional sum of orthonormal spline wavelet systems, such as the Battle–Lemarié wavelets. We also consider the natural extensions to Triebel–Lizorkin spaces. This builds upon, and is a generalization of, previous work of Seeger and Ullrich, where analogous results were established for the Haar wavelet system.