; TeX output 1998.07.20:1703 G!Í /;O5f cmcsc8Doc.MauUth.J.DMV K`y cmr10713*(+؍ K@r cccsc10@DevelopmentsfromNonharmonicFfourierSeries썒 f+- cmcsc10KristianSeip!_';;OAbstract.]W*e~bGeginthissurveybyshowingthatPaleyandWiener'sun- ;;Oconditional3basisproblemfornonharmonicF*ourierseriescanbGeunderstood;;OasPaproblemabGoutweightedPnorminequalitiesforHilbertoperators. Then;;Owereformulatethebasisprobleminamoregeneralsetting,Yanddiscuss;;OBeurling-typGedensitytheoremsforsamplingandinterpGolation.^Next,we;;Ostateusomemultipliertheorems,ofasimilarnatureasthefamousBeurling-;;OMalliavin@theorem,;andsketchtheirroleinthesub 8ject.aFinally*,;wediscuss;;OextensionsofnonharmonicF*ourierseriestoweightedPaley-Wienerspaces,;;OandindicatehowthesespacesareexploredviadeBranges'HilbGertspacesof;;OentireUUfunctions.v;;O1991UUMathematicsSub 8jectClassication:q30,42,46Rk1.FromLP aley-WienertoHunt-Muckenhoupt-Wheeden /;OThe@theoryofnonharmonicF*ourierseriesbGeginswithPaleyandWiener[18], /;Owhodiscoveredthatthetrigonometricsystem!", cmsy10fb> cmmi10e^ 0er cmmi7ik+BxHgremainsanunconditional/;ObasisforL^ٓR cmr72|s( [;)whentheintegerfrequencieskEarereplacedby\nonharmonic"/;Ofrequenciespk\rsatisfyingjk6η K>kPjdpforsomed<1=[ٟ^2L.mThispresultledtoquite/;OextensiveCactivityaroundtheproblemofdescribingallunconditionalbasesofthe/;OformQfe^iO \ cmmi5k_xgforL^2|s( [;).YA>decisiveQbreaktroughwasmadebyPavlov.[19],and/;OaUUcompletesolutiontotheproblemasjuststatedisnowavqailable[9,12,15].;;OW*e>shallpresentbGelowasurveyofrecentdevelopmentswhicharecloselyre-/;Olated%totheproblemofPaleyandWiener.8LetusthereforebGeginbyclarifying/;OhowXtheunconditionalbasisproblemcanbGeunderstood:xItcanberecastasa/;OquestionconcerningbGoundednessofHilbertoperatorsincertainweightedL^2.(or/;Omore.SgenerallyL^pR)spacesoffunctionsandsequences,dandthusleadsustothe/;OHunt-Muckenhoupt-Wheedenntheorem[7].W*ewillfollowC[12],uinwhichthisshift/;OfromUUHilbGertspacegeometrytoweightedUUnorminequalitiesismade.;;OW*e?restatethePaley-Wienerproblemintermsofentirefunctions.yDenoteby/;OPcW^p (0