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FunctionsofboundedJordanvariation
17
Geometrically,
ζe
increaseslinearlyby
d1
ontheinterval
[
0,
c1]
sothat
ζe(c1)=d1
.Afterwards
ζe
decreaseslinearlyby
d2
on
[c1
,
c1+c2]
,
increaseslinearlyby
d3
on
[c1+c2
,
c1+c2+c3]
,decreaseslinearlyby
d4
on
[c1+c2+c3
,
c1+c2+c3+c4]
,andsoon.Forthisreasonwe
call(1.8)azigzagfunctionoforderó.
Itfollowsfromtheconstructionandcontinuityofthezigzagfunc-
tion
ζe
that
ζe(
1
)<+∞
forall
ó>
0.Again,itisilluminatingto
determineallvaluesof
ó>
0forwhichthezigzagfunction
(1.8)
be-
longstothefunctionclassesintroducedsofar.Ofcourse,thefunction
ζe
isalwayscontinuous,byconstruction,butnotdifferentiableatits
peaks.Soitisonlyinterestingtofindall
ó>
0forwhich
ζe
isHölder
(inparticular,Lipschitz)continuous.Choosing
cn
and
dn
asin
(1.7)
and0<γ<1weobtain
sup{dnc
−γ
n
|
|n=1,2,...}=sup{n
−e2nγ|
|n=1,2,...}=+∞,
sincetheexponentialterm2
nγ
growsessentiallyfasterthanthepower-
typeterm
ne
.Sowegetthesomewhatdisappointingresultthat,
looselyspeaking,thezigzagfunction
ζe
doesnot“feel”thedepen-
denceon
ó
,astheoscillationfunction
ωα,β
feelsthedependence
on
α
and
β
.Wesummarizethebehaviorof
ζe
withthefollowing
proposition.
101040Proposition0
Thezigzagfunction
(1.8)
belongsto
C
forallvaluesof
ó>0,butdoesnotbelongtoLipγforanyγ∈(0,1].
Theoscillationfunction
(1.6)
andthezigzagfunction
(1.8)
are
usefulforconstructingcounterexamples.Forfurtherreference,we
collectintheTable1.1thevaluesof
α
,
β
,and
ó
,respectively,forwhich
thesefunctionsbelongtosomeofthefunctionspacesoccurringin
(1.5)
.
AnessentialextensionofthistablewillbegiveninTables1.3
and1.4attheendofthischapter.
Letusnowrecallthedefinitionoftheclassicalspace
BV
which,
asfarasweknow,goesbacktoCamilleJordan[24]andisourmain
objectofattentioninthissurvey.Throughoutthefollowing,wedenote
by
P
thefamilyofallpartitionsoftheinterval
[
0,1
]
,i.e.,allfinitesets
P={t0,t1,...,tm−1,tm},wherem∈Nisvariable,satisfying(1.1).
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