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can be used to match
f
1
,
f
2
,
f
2
through the three free constants
c
0
,
c
2
,
c
4
.Inthe
odd case,
c
3
z
3
c
5
z
5
s
(
z
)
=
c
1
z
+
+
(1.594)
can be used to match
f
1
,
f
2
,
f
2
through the three free constants
c
1
,
c
3
,
c
5
.The
canonical Hermite cubics are replaced by canonical quadratics defined on the inter-
val (
−
1,1) of the ξ-axis by
⎩
L
2
(
3ξ
−
1
)(
ξ
+
1
)
,
−
η
0
(ξ)
=−
3ξ
−
2ξ
+
1
=−
1
≤
ξ
≤
0
R
2
1)
2
η
0
(ξ)
=
η
0
(ξ)
=
ξ
−
2ξ
+
1
=
(ξ
−
0
≤
ξ
≤
1
(1.595)
0,
otherwise,
⎩
0
(ξ)
2
ψ
=
ξ
+
ξ
=
ξ(ξ
+
1),
−
1
≤
ξ
≤
0
R
ψ
0
(ξ)
=
=
≤
ξ
≤
(1.596)
ψ
0
(ξ)
0,
0
1
0,
otherwise.
In the even case, we can write
R
L
f
2
ψ
L
s
(
z
)
=
f
1
η
1
(
z
)
+
f
2
η
2
(
z
)
+
2
(
z
).
(1.597)
The supporting polynomials are quartics that can be defined in terms of the canon-
ical quartics by
R
R
L
L
L
η
1
(
z
)
=
η
0
(ζ
1
), η
2
(
z
)
=
η
0
(ζ
2
)
+
2ψ
0
(ζ
2
),
(1.598)
z
2
2
ψ
L
L
ψ
2
(
z
)
=
0
(ζ
2
),
(1.599)
with
z
2
z
2
z
2
,
z
2
z
2
,
2
−
ζ
1
=
=
(1.600)
2
z
z
2
.
ζ
1
(
z
)
=
ζ
2
(
z
)
=
(1.601)
The termζ
1
maps the interval (0,
z
2
) onto the interval (0,1) of the ζ-axis, while ζ
2
maps the interval (0,
z
2
) onto the interval (
−
1,0) of the ζ-axis. The corresponding
derivatives are
η
1
(
z
)
η
0
(ζ
2
)
2ψ
0
(ζ
2
)
,
2
z
2
z
z
2
z
2
η
0
(ζ
1
), η
2
(
z
)
=
=
+
(1.602)
z
ψ
2
(
z
)
z
2
ψ
0
(ζ
2
).
=
(1.603)
In the odd case, we can write
f
1
ψ
R
L
f
2
ψ
L
s
(
z
)
=
1
(
z
)
+
f
2
η
2
(
z
)
+
2
(
z
).
(1.604)
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