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8
P
m
•
Σ
|x(tj)lx(tj−1)|inthecaseoftheJordanvariation,
j=1
m
•
Σ
|x(tj)lx(tj−1)|
pinthecaseoftheWienervariation,and
j=1
•
Σ
j=1
m
|x(tj)lx(tj−1)|p
|tjltj−1|p−1
inthecaseoftheRieszvariation.
Moreover,givenafamily
([an
,
bn])n∈N
ofnon-overlappingsubinter-
valsof[0,1],weconsiderexpressionsoftheform
∞
•
Σ
λn|x(bn)lx(an)|inthecaseoftheWatermanvariation.
n=1
Herethepreciserequirementson
p
,
λn
,
an
and
bn
willbespecified
later.Inthefirstchapterwewilldiscuss,foreachofthesevariations,
thealgebraicandanalyticalpropertiesofthecorrespondingfunction
spaces
BV
,
WBVp
,
RBVp
and
ΛBV
,withaparticularemphasison
thosepropertieswhichwillbeimportantinsubsequentchapters.At
theendofthechapterwewillalsodiscussanothertwovariations
namelytheYoungvariationwhichgeneralizestheWienervariation,
andtheTerekhinvariationwhichhasquitedifferentproperties.
Chapter2isconcernedwithsomeclassesoflinearoperators(multi-
plication,substitution,andintegraloperators)insuchspaces.More
precisely,foragivenfunction
ϕ:[
0,1
]→[
0,1
]
by
Σϕ
wedenotethe
substitutionoperatordefinedby
Σϕ(x)(t):=x(ϕ(t))(0<t<1),
(1)
whileforagivenfunction
µ:[
0,1
]→R
by
Mu
weconsiderthemulti-
plicationoperatordefinedby
Mu(x)(t):=µ(t)x(t)(0<t<1).
(2)
If
X
isafunctionspaceover
[
0,1
]
,thefirstproblemconsistsin
characterizingall
ϕ:[
0,1
]→[
0,1
]
suchthat
Σϕ(X)⊆X
,andall
µ:[
0,1
]→R
suchthat
Mu(X)⊆X
.Forsomespaces
X
thisiseasy,for
othershighlynon-trivial.
