Continuity properties of best analytic approximati

arXiv:math/9511214v1[math.FA]28Nov1995CONTINUITYPROPERTIESOFBESTANALYTICAPPROXIMATIONV.V.PELLERANDN.J.YOUNGAbstract.LetAbetheoperatorwhichassignstoeachm×nmatrix-valuedfunc-tionontheunitcirclewithentriesinH∞+Citsuniquesuperoptimalapproximantinthespaceofboundedanalyticm×nmatrix-valuedfunctionsintheopenunitdisc.WestudythecontinuityofAwithrespecttovariousnorms.Ourmainresultisthat,foraclassofnormssatifyingcertainnaturalaxioms,Aiscontinuousatanyfunctionwhosesuperoptimalsingularvaluesarenon-zeroandissuchthatcertainassociatedintegerindicesareequalto1.WealsoobtainnecessaryconditionsforcontinuityofAatpointandasufficientconditionforthecontinuityofsuperoptimalsingularvalues.1.IntroductionTheproblemoffindingabestuniformapproximationofagivenboundedfunctionontheunitcirclebyananalyticfunctionintheunitdiscisanaturalonefromtheviewpointofpuremathematicsanditalsohasengineeringapplications,forexampleinH∞control[F],broadbandimpedancematching[He]androbustidentification[Par].Inthesecontexts,toeffectadesignorconstructamodel,onemustcomputesuchabestapproximation,andinorderthatnumericalcomputationshavevalidityitisimportantthatthesolutiontobecomputeddependcontinuouslyontheinputdata,forotherwisetheimperfectprecisionoffloatingpointarithmeticmayleadtohighlyinaccurateresults.Itisthereforesomewhatdisconcertingthat,withrespecttotheL∞norm,theoperatorofbestanalyticapproximationisdiscontinuouseverywhereexceptatpointsofH∞[M,Pa].Neverthelessengineersregularlycomputesuchapproximationsandappeartofindtheresultsreliable.AwaytoaccountforthiswouldbetoshowthatbestanalyticapproximationiscontinuousonsuitableBanachsubspacesofL∞(T)withnormswhichmajorisetheuniformnorm,oratleast,iscontinuousatmostpointsofthespace.Onecanexpectthatmostfunctionsofengineeringinterestwilllieinoneofthesewell-behavedsubspaces,andthattheerrorsDate:February1,2008.1991MathematicsSubjectClassification.AMSSubjectClassifications:30E10,47B35,93B36.V.V.Peller’sresearchwassupportedbyanNSFgrantinModernAnalysis.N.J.YoungwishestothanktheMathematicsDepartmentsofKansasStateUniversityandtheUniversityofCaliforniaatSanDiego,andtheMathematicalSciencesResearchInstituteforhospitalitywhilethisworkwascarriedout.ResearchatMSRIissupportedinpartbyNSFgrantDMS-9022140.12V.V.PELLERANDN.J.YOUNGintroducedbycomputerarithmeticwillresultinperturbationswhicharesmallintheassociatednorm.WearethusledtoaskforwhichBanachspacesX⊂L∞(T)theoperatorAofbestanalyticapproximationmapsXintoXandiscontinuousatagenericpointofX(insomesense).Thisquestionhasbeenthoroughlyanalysedforthecaseofscalar-valuedfunctions.Itwasshownin[P1]that,forspacesX⊂H∞+Csatisfyingsomenaturalaxioms,therestrictionofAtoXiscontinuouswithrespecttothenormofXatafunctionϕifandonlyifkHϕkisasimplesingularvalueoftheHankeloperatorHϕ.Analogousquestionsformatrix-valuedfunctionsarealsoofinterest,particularlyfortheirrelevancetoengineeringapplications.Theyareagooddealmorecomplicatedthaninthescalarcase.Tobeginwith,thereistypicallynouniquebestanalyticapproximationinthematrixcase,whenwemeasureclosenessbytheL∞norm.Inordertospecifyanapproximationuniquelyandsoobtainawellformulatedquestionofcontinuitywecanuseamorestringentcriterionofapproximation.Thenotionofasuperoptimalapproximationisanaturaloneformatrix-valuedfunctions:byimposingtheconditionoftheminimisationofthesupremaofallsingularvaluesoftheerrorfunctionitgivesauniquebestapproximantinmanycases.Hereisaprecisedefinition.DenotebyMm,nthespaceofm×ncomplexmatricesendowedwiththeoperatornormasaspaceoflinearoperatorsfromCntoCmwiththeirstandardinnerproducts.LetH∞(Mm,n)denotethespaceofboundedanalyticMm,n-valuedfunctionsontheunitdiscDwithsupremumnorm:||Q||H∞def=||Q||∞def=supz∈D||Q(z)||Mm,n.Similarly,L∞(Mm,n)denotesthespaceofessentiallyboundedLebesguemeasurableMm,n-valuedfunctionsonTwithessentialsupremumnorm.ByFatou’stheorem[H,p.34]functionsinH∞(Mm,n)haveradiallimitsa.e.onT,sothatH∞(Mm,n)canbeembeddedisometricallyinL∞(Mm,n),andweshalloftentacitlyregardelementsofH∞(Mm,n)asfunctionsontheunitcircle.WherethereisnoriskofconfusionweshallsometimeswriteH∞,L∞forH∞(Mm,n),L∞(Mm,n).WedefineH∞+Ctobethespaceof(matrix-valued)functionsonTwhichareexpressibleasthesumofanH∞functionandacontinuousfunctiononT.ForanymatrixAwedenotethetransposeofAbyAtandthesingularvaluesors-numbersofAbys0(A)≥s1(A)≥···≥0.ForF∈L∞(Mm,n)wedefine,forj=0,1,2,...,s∞j(F)def=esssup|z|=1sj(F(z))ands∞(F)def=(s∞0(F),s∞1(F),s∞2(F),...).CONTINUITYPROPERTIESOFBESTANALYTICAPPROXIMATION3WeshallsaythatQ∈H∞(Mm,n)isasuperoptimalH∞approximanttoΦ∈L∞(Mm,n)ifs∞(Φ−Q)isaminimumoverQ∈H∞withrespecttothelexico-graphicordering.Itwasprovedin[PY1]thatifanm×nmatrixfunctionΦisinH∞+CthenthereisauniquesuperoptimalapproximanttoΦinH∞(Mm,n).Weshalldenotethisap-proximantbyAΦ.In[PY1],inadditiontoprovinguniqueness,weobtaineddetailedstructuralinformationaboutthe“superoptimalerror”Φ−AΦandweestablishedseveralheredityresults(thatis,theoremsoftheform“Φ∈XimpliesAΦ∈X”forvariousfunctionspacesX).InanyspacewhichdoeshavethishereditypropertyitisnaturaltoaskwhetherAactscontinuously.Weshallshowthatforasubstantialclassofnormstherearemanycontinuityp