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BV-typespaces1
1
Inthischapterwearegoingtocollectallnotionsandfactson
BV
-type
spacesweneedinthesequel.Withoutlossofgenerality,wewillrestrict
ourselvestofunctions
x:[
0,1
]→R
;everyresulteasilycarriesoverto
functionson
[a
,
b]
throughthelinearisomorphisms
tl→a+(bla)t
whichrespects,uptoaconstant,thefinitenessofeveryvariationwe
aregoingtodefine.
Wewillbasicallybeworkinginthesettingoffourtypesofvariation.
GivenapartitionP={t0,t1,...,tm−1,tm},wherem∈N,suchthat
0=t0<t1<...<tm−1<tm=1,
(1.1)
weconsiderexpressionsoftheform
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-overlapping
1
subinter-
valsof[0,1],weconsiderexpressionsoftheform
∞
•
Σ
λn|x(bn)lx(an)|inthecaseoftheWatermanvariation.
n=1
1
Twointervalsarecallednon-overlapping,iftheirunioncontainsatmostone
point.
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