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1、ForTheresa,Sophie,Alexandra,ennifer,andErikarefa
eolynomialspervademathemati
s,andmu
hthatisbeautifulinmathe-mati
sisrelatedtopolynomials.Virtuallyeverybran
hofmathemati
s,fromalgebrai
numbertheoryandalgebrai
geometrytoappliedanaly-sis,Fourieranalysis,and
2、
omputers
ien
e,hasits
orpusoftheoryarisingfromthestudyofpolynomials.Histori
ally,questionsrelatingtopolyno-mials,forexample,thesolutionofpolynomialequations,gaverisetosomeofthemostimportantproblemsoftheday.Thesubje
tisnowmu
htoolargetoattemptanen
y
lopedi
3、overage.Thebodyofmaterialwehoosetoexplore
on
ernsprimarilypolyno-mialsastheyariseinanalysis,andthete
hniquesofthebookareprimarilyanalyti
.Whilethe
onne
tingthreadisthepolynomial,thisisananalysisbook.Thepolynomialsandrationalfun
tionsweare
on
ernedwitharealmo
4、stex
lusivelyofasinglevariable.Weassumeatmostaseniorundergraduatefamiliaritywithrealand
omplexanalysis(indeedinmostpla
esmu
hlessisrequired).However,thematerialisoftenterselypresented,withmu
hmathemati
sexploredintheexer
ises,someofwhi
harequitehard,manyofwh
5、i
haresuppliedwith
opioushints,somewith
ompleteproofs.Welloverhalfthematerialinthebookispresentedintheexer
ises.Thereaderisen
ouragedtoatleastbrowsethroughthese.Wehavebeenmu
hin
uen
edbyolyaandSzeg}o's
lassi
roblemsandTheoremsinAnalysis"inourapproa
htoth
6、eexer
ises.(ThoughunlikeolyaandSzeg}owehosetoin
orporatethehintswiththeexer
ises.)viiirefa
eThebookismostlyself-
ontained.Thetext,withouttheexer
ises,pro-videsanintrodu
tiontothematerial,butmu
hoftheri
hnessisreservedfortheexer
ises.Wehaveattemptedtohighl
7、ightthepartsofthetheoryandthete
hniqueswendmostattra
tive.So,forexample,untz'slovelyhara
terizationofwhenthespanofasetofmonomialsisdenseisexploredinsomedetail.Thisresultepitomizesthebestofthesubje
t:anattra
tiveandnontrivialresultwithseveralattra
tiveandno
8、ntrivialproofs.Thereareex
ellentbooksonorthogonalpolynomials,Chebyshevpoly-nomials,Chebyshevsystems,andthegeometryofpolynomials,tonamebutafewofthetopi
swe
over,anditisnotourintenttorewriteanyoft