關(guān)于Hlder正則性的一個(gè)注記

胡琳,邢春峰,李林杉

(北京聯(lián)合大學(xué)基礎(chǔ)部,北京 100101)

胡琳,邢春峰,李林杉

(北京聯(lián)合大學(xué)基礎(chǔ)部,北京 100101)

Hlder正則性在圖像消噪、信號重構(gòu)等方面都具有重要意義.本文對Hlder空間的兩種定義進(jìn)行研究分析,利用歸納法和Littlewood-Paley分解方法證明了兩種Hlder定義在一定條件下還是等價(jià)的,為Hlder正則性的應(yīng)用提供了理論依據(jù).

Hlder空間;Littlewood-Paley分解;等價(jià)性

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

1 引言

正則性一般用來表征函數(shù)的光滑程度,在圖像消噪、信號重構(gòu)等方面都具有重要意義[15].文獻(xiàn)[6-7]給出Hlder正則性的經(jīng)典定義,文獻(xiàn)[8-10]利用正交小波刻劃了函數(shù)的Hlder正則性.文獻(xiàn)[11-12]分別利用切波變換和曲波變換研究了函數(shù)的Hlder正則性,并利用多項(xiàng)式函數(shù)給出Hlder正則性的另一定義.本文利用歸納法和Littlewood-Paley分解方法證明了兩種Hlder定義的等價(jià)性.

2 預(yù)備知識

3 主要結(jié)果

參考文獻(xiàn)

[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]龍瑞麟.高維小波分析[M].北京:世界圖書出版社,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.

國家自然科學(xué)基金數(shù)學(xué)天元基金(11226156);北京聯(lián)合大學(xué)新起點(diǎn)項(xiàng)目(Zk10201308).

胡琳(1979-),博士,講師,研究方向：小波分析.

2010 MSC:42C15,42B10,26B35