applications of conjugate operators to determination of jumps for functions

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1、ActaMath.Hungar.,134(4)(2012),439–451DOI:10.1007/s10474-011-0131-1FirstpublishedonlineJune17,2011APPLICATIONSOFCONJUGATEOPERATORSTODETERMINATIONOFJUMPSFORFUNCTIONS∗X.L.SHIandW.WANGCollegeofMathematicsandComputerScience,HunanNormalUniversity,Changsha410081,Chinae-mails:xianliangshi@yahoo.co

2、m,wwang.wei011@yahoo.com.cn(ReceivedDecember25,2010;acceptedFebruary14,2011)Abstract.Weprovethattheconjugateconvolutionoperatorscanbeusedtocalculatejumpsforfunctions.OurresultsgeneralizethetheoremsestablishedbyHeandShi.Furthermore,byusingLuk´acsandM´oricz’sidea,wesolveanopenquestionposedby

3、ShiandHu.1.IntroductionDeterminationofjumpsforfunctionsplaysanimportantroleindetec-tionofedges(see[4]).Manyauthors,suchasL.Fej´er,F.Luk´acs,B.L.Golubov,G.Kvernadge,F.M´oricz,A.Gelb,E.Tadmor,X.L.Shi,Q.L.Shi,L.Hu,H.Y.Zhang,P.Zhou,S.P.Zhou,Z.T.He,etc.wereinterestedinthisquestion(see[3–17]).Re

4、cently,Z.T.HeandX.L.Shi[6]introducedanewmethodtodeterminejumpsforfunctions.LetφbeafunctioninL(R)withφ(x)dx=1.DenotebyφtheHilberttransformofφ,i.e.R1φ(x−y)1φ(x−y)φ(x):=H(φ)(x):=p.v.dy:=limdy.πRyπε→0+

5、y

6、>εyForthefollowingthreespecialkernels:1−x2(1.1)φ(x)=√e,π11(1.2)φ(x)=,π1+x2∗Correspond

7、ingauthor.Keywordsandphrases:Hilberttransform,convolution,jump,Shannonwavelet.2000MathematicsSubjectClassification:42A50,42A16.0236-5294/$20.00c2011Akad´emiaiKiad´o,Budapest,Hungary440X.L.SHIandW.WANGand−1∞dt1(1.3)φ(x)=2,01+t41+x4theyprovedthatiff∈L(R)andξ∈Risasimplediscontinuityoff,the

8、nπlim−Tn(f)(ξ)=dξ(f),n→∞lnnwhere(1.4)dξ(f)=f(ξ+0)−f(ξ−0),andTn(f)isdefinedasfollows.Setφn(x)=nφ(nx),then(1.5)Tn(f)(x):=φn∗f(x).Thefirstaimofthispaperistogeneralizetheirresultstomoregeneralcases.Theorem1.Assumethatthefunctionφ∈L(R)satisfiesthefollowingconditions:i)φ(x)dx=1;Rii)φiseven,i

9、.e.φ(−x)=φ(x);Ciii)φ(x)1+α,α>0;1+

10、x

11、Cω

12、u

13、iv)φ(x+u)−φ(x−u)1+α,α>0,1+

14、x

15、whereω(t)isamodulosofcontinuitywhichsatisfies1ω(t)dt<∞.0tIfξ∈Risasimplediscontinuityoff∈L1(R),thenπ(1.6)lim−Tn(f)(ξ)=dξ(f),n→∞lnnwhereTnisdefinedasin(1.5).ActaMathematicaHungar