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Adding these inequalities together gives an inequality of the type (
B.30
), where
p
m
+
q
m
=
f
k,m
+
2
−
f
k,m
+
1
,
w
m
=
u
k,
m
+
1
−
≥
u
k,m
,m
1. We apply Lemma
B.3.9
to
w
k
(t)
defined in (
B.52
), with
s(t)
=
f
k
(t
+
k)
and
w
0
:=
u
k,
1
−
Pu
0
. Observing
kU
k
(t)
, we get (
B.61
) with (
B.60
) from (
B.53
), (
B.54
).
that
w
k
(t)
=
From (
B.61
) the size of
E(k,T )
as
k
→
0 for fixed
T>
0 is important. We have
Lemma B.3.14
Assume
0
,u
0
∈
K
·
H
and
(
B.11
).
Then
E(k,T )
=
o(
1
) as k
→
0
,
(B.62a)
and
,
for compatible data satisfying
(
B.15a
), (
B.15b
),
it holds
E(k,T )
≤
Ckas k
→
0
.
(B.62b)
Proof
We show (
B.62b
). The terms in the second row of the definition (
B.60
)of
E(k,T )
can be bounded as in (
B.62b
), by (
B.26
), (
B.28
). The terms in the first row
of (
B.60
) are boun
d u
sing
(
B.15a
), (
B.15b
) as follows: we decompose
f
=
g
+
h
∈
S(
0
,T)
and define
g
k
(t), h
k
(t)
as in (
B.55
). Assume
k
=
T/M
for
M
∈ N
. Then
h(t
+
k)
−
h(t)
L
2
(J
;
V
)
≤
k
h
(t)
L
2
(
0
,T
+
k
;
V
∗
)
,
h
k
(t)
−
h(t)
L
2
(J
;
V
)
≤
k
h
(t)
L
2
(
0
,T
+
k
;
V
∗
)
,
as well as
M
−
1
g(t
+
k)
−
g(t)
L
2
(J
;
H
)
=
J
k,m
g(τ
+
k)
−
g(τ)
H
d
τ
m
=
0
M
−
1
k
=
0
g(τ
+
(m
+
1
)k)
−
g(τ
+
mk)
H
d
τ
0
m
=
M
−
1
≤
k
Va r
(g, J
k,m
;
H
)
m
=
0
≤
;
H
k
Va r
(g, J
).
Furthermore,
g(τ)
d
τ
−
g(t)
H
M
−
1
1
k
g
k
(t)
−
g(t)
L
1
(J
;
H
)
=
d
t
J
k,m
J
k,m
m
=
0
M
−
1
1
k
≤
J
k,m
g(s)
−
g(t)
H
d
t
d
s
J
k,m
m
=
0
M
−
1
≤
k
Va r
(g, J
k,m
;
H
)
m
=
0
≤
k
Va r
(g, J
;
H
).
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