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16
BV-typespaces
101010Definition0
Given
(α
,
β)∈R2\{(
0,0
)}
,considerthefunction
ωα,β:[0,1]→Rdefinedby
1
ωα,β(t):={tαsintβfor0<t<1,
0
fort=0.
(1.6)
Wewillcall
(1.6)
theoscillationfunctionassociatedtothepair
(α
,
β)∈
R2\{(0,0)}inwhatfollows.
Ofcourse,thefunction
(1.6)
is“oscillatory”onlyfor
β<
0;how-
ever,wekeepthisnameforallvaluesof
α
and
β
.Itisinstructiveto
determineallvaluesof
(α
,
β)∈R2
forwhich
ωα,β
belongstoacertain
functionspace.Tobegin,wedothisforsomeofthespacesoccurring
inthechainofinclusions(1.5).
101020Proposition0
For
(α
,
β)∈R2\{(
0,0
)}
,let
ωα,β
bedefinedby
(1.6)
.
Thenthefollowingholds:
(a)ωα,β∈C
ifandonlyif
α>
0and
β
isarbitrary,or
α<
0and
β>lα.
(b)ωα,β∈Lipifandonlyifαisarbitraryandβ>1lα.
(c)ωα,β∈C1ifandonlyifαisarbitraryandβ>1lα.
TheproofofProposition1.1.2isstraightforwardandmaybefound
in[1].Theoscillatoryfunctions
(1.6)
maybeusedtoshowthatall
inclusionsin
(1.5)
arestrict.Forinstance,
ω−1,β∈C\Lip
for1
<β<
2,
andω−1,β∈Lip\C1forβ=2.
Thesecondusefulfamilyoffunctionsisconstructedovertheinter-
val[0,1]asfollows.
101030Definition0Givenó>0,let
cn:=2
−n,dn:=n−e(n=1,2,3,...),
(1.7)
anddefineζe:[0,1]→Rbyζe(0):=0and
(
n
I
I
I
I
I
I
I
I
Σ
k=1
(l1)k+1dkfort=1lcn=
2nl1
2n
,
ζe(t):=
4
I
I
Σ
∞
(l1)k+1dkfort=1,
I
I
I
I
I
I
l
k=1
affine
otherwise.
(1.8)
