一类新的Haar子空间中最佳逼近的唯一性-论文.pdf

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1、第35卷第4期吉首大学学报(自然科学版)Vo1．35No．42014年7月JournalofJishouUniversity(NaturalScienceEdition)Ju1．20l4ArticleID：1007—2985(2014)04—0008—05UniquenessoftheBestApproximationinaNewHaarTypeSpaceFANGDonghui(CollegeofMathematicsandStatistics，JishouUniversity，Jishou416

2、000，HunanChina)Abstract：Theproblemofthebestapproximationwithgeneralizedrestrictionsisconsideredinthispaper．ByintroducinganewHaartypespace，theuniquenessandstronguniquenessofthebestapproximationonthisHaartypespacearegiven．Keywords：bestapproximation；Haar

3、typespace；uniqueness；stronguniquenessCLCnumber：0174．41Documentcode：ADOI：10．3969／j．issn．1007—2985．2014．04．0021IntroductionIetubean一dimensionalsubspaceofCEa，b]，thespaceofreal—valuedcontinuousfunctionswithL。。norm，thatis，I厂fl—maxIf(x)I，Vf∈CEa，6]．LetL：UC(K

4、)bealinearoperatormap—pingUintoC(K)，whereKisafiniteunionofintervalsandpointsinR．Forgivenl，u∈C(K)，setU(Z，)={P∈U：Z(￡)≤Lp()≤“(￡)，tEK}，whereZislowersemicontinuousand“1suppersere一continuousonK．Inthispaper，wealwaysassumethatU(Z，)≠．Nowweconsidertheproblemofa

5、pproximatinginnormz∞byelementsofU(z，“)．Foragivenfunctionf∈CEa，6]，wesaythatP0EUn(￡，M)isabestapproximationofffromU(z，乱)，if【l厂一P。li—inf{Il厂一P：P∈U(Z，“))．Theuniquenessandstronguniquenessofthebestapproximationhavereceedmuchattention(ref．E1—9])．Inref．[1]，Cha

6、lmersandTaylorintroducedageneralmethodofinvestigatingunlquenesolbestapproximationswithconstraints，whichprovidedaunifiedapproachtotheproblem·However，thisaDproachessentiallyprovidedonlysufficientconditionsforuniquenessofbestconstrainedaPProxma—tion．Inre

7、f．[43，theauthorsintroducedanewHaarspace，namely，L—Haarspace，andestablishedaHaa一typetheoryforconstrainedapproximation，andgavesomenecessaryandsufficientconditionstorumque—nessandstronguniqueness．Notethat工一HaarisnotHaarasshowninref．[9-1．Thus，theauthorsinr

8、ef·r9]introducedanotherHaartypespaceandstudiedthecharacterizationforthisHaartypespa·Inspiredbythoseworksmentionedabove，wecontinuetostudytheproblemofthebestapp0xm一tionwithgeneralizedrestrictions．ByintroducinganewHaartypespacedefinedinref．E9-1'w