关于Hlder正则性的一个注记

胡琳,邢春峰,李林杉

(北京联合大学基础部,北京 100101)

胡琳,邢春峰,李林杉

(北京联合大学基础部,北京 100101)

Hlder正则性在图像消噪、信号重构等方面都具有重要意义.本文对Hlder空间的两种定义进行研究分析,利用归纳法和Littlewood-Paley分解方法证明了两种Hlder定义在一定条件下还是等价的,为Hlder正则性的应用提供了理论依据.

Hlder空间;Littlewood-Paley分解;等价性

DO I：10.3969/j.issn.1008-5513.2013.05.006

1 引言

正则性一般用来表征函数的光滑程度,在图像消噪、信号重构等方面都具有重要意义[15].文献[6-7]给出Hlder正则性的经典定义,文献[8-10]利用正交小波刻划了函数的Hlder正则性.文献[11-12]分别利用切波变换和曲波变换研究了函数的Hlder正则性,并利用多项式函数给出Hlder正则性的另一定义.本文利用归纳法和Littlewood-Paley分解方法证明了两种Hlder定义的等价性.

2 预备知识

3 主要结果

参考文献

[1]M allat S,Hwang W L.Singu larity detection and p rocessing w ith w avelets[J].IEEE T rans.Inf.Theory., 1992,38(2):617-643.

[2]Rantala M,V¨ansk¨a S,J¨arvenp¨a¨a S,et al.Wavelet-based reconstruction for Lim ited angel X-ray tomography[J].IEE Trans.Med.Imag.,2006,25(2):210-217.

[4]V´ehel J L.Signal enhancement based on Hlder regu larity analysis[J].IMA Volumes in Mathematics and its app lications,2002,132:197-209.

[5]Cannarsa P,Cardaliaguet P,G iorgieri E.Hlder regu larity of the norm al d istance w ith an app lication to a PDE model for grow ing sandpiles[J].Transactions of the American M athematical Society,2007,359(6): 2741-2775.

[6]Jaf ard S.Pointw ise sm oothness,tw o-m icrolocalization and w avelet coef cients[J].Pub l.M at.Barc.,1991, 35(1):155-168.

[7]Holschneider M,Tcham itchian P.Pointw ise analysis of Riemann′s nondif erentiable function[J].Invent. M ath.,1991,105(1):157-175.

[8]M eyer Y.W avelets and Operators[M].Cam bridge:Cam bridge University Press,1992.

[9]Daubechies I.Ten Lectures on W avelets[M].Philadelphia:Society for Industrial and App lied Mathematics (SIAM),1992.

[10]龙瑞麟.高维小波分析[M].北京:世界图书出版社,1995.

[11]LakhonchaiP,Sum etkijakan S,Sam po J.Shearlet transform sand directional regularities[J].Int.J.W avelets M ultiresolut.In f.Process.,2010,8(5):743-771.

[12]Nualtong K,Sumetkijakan S.Analysis of Hlder regu larity by wavelet-like transform sw ith parabolic scaling[J].Thai J.M ath.,2005,3(2):275-283.

Hu Lin,Xing Chunfeng,Li Linshan

(Departm ent of Basic Cou rses,Beijing Union University,Beijing 100101,China)

Hlder regu larity p lays an im portan t part in various signal processing app lications such as denoising and reconstruction.W ith the help of Littlewood-Paley decom position technique,equivalence of two defnitions for Hlder space is proved which is the theoretical basis for the app lication of Hlder regularity.

Hlder space,Littlewood-Paley decom position,equivalence

O 174.2

A

1008-5513(2013)05-0477-07

2013-07-06.

国家自然科学基金数学天元基金(11226156);北京联合大学新起点项目(Zk10201308).

胡琳(1979-),博士,讲师,研究方向：小波分析.

2010 MSC:42C15,42B10,26B35