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Summary: UniquenessofRepresentationbyTrigonometricSeries
J.MARSHALL ASH', DePaul University,Chicago,IL 60614
J.MARSHALLASH receivedhisS.B.,S.M.,andPh.D.degreesattheUniver- I"k
sityofChicagowherehewasa studentofAntoniZygmund.Hewasa Joseph
FelsRittinstructoratColumbiaUniversityfrom1966to1969.Sincethenhe
hasbeenontheDePaulUniversityfaculty,wherehebecamea fullprofessor
in1974andchairedfrom1985to1987.In 1977hewasvisitingprofessorat
StanfordUniversity.His researchinterestshave centeredon multiple
trigonometricseries,singularintegrals,andrealanalysis,withforaysinto
functionalandnumericalanalysis.
Abstract.In 1870GeorgCantorprovedthata 2srperiodiccomplexvaluedfunctionofa realvariable
coincideswiththevaluesofatmostonetrigonometricseries.Wepresenthisproofandthensurveysome
ofthemanyonedimensionalgeneralizationsandextensionsofCantor'stheorem.We alsosurveythe
situationinhigherdimensions,wherea greatdeallessisknown.
1. Cantor'suniquenesstheorem.In 1870Cantorproved
THEOREMC (Cantor[5]). If, foreveryrealnumberx
N
lim E ce inx = 0,
N-?oo n=-N
thenall thecomplexnumberscn, n = 0,1, - 1,2, - 2,... arezero.
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