Pseudo-spectral derivative of quadratic quasi-interpolant splines

358 S. Remogna
Therefore
(19)
From (19) and (17), we get (18).
h p
δ p
≤ A 2 .
where
Theorem 1 leads immediately to the following global results.
THEOREM 2. Let f ∈ C r (I), r≤ 2. Then
∥ ( f − Q2 f) ′∥ ∥
∞
≤ C r h r−1 ω( f (r) ,h,I),
C r = 30(5/2)r+1
r!
h
δ .
If in addition f ∈ C 3 (I), then
∥ ( f − Q2 f) ′∥ ∥
∞
≤ C 2 h 2∥ ∥ ∥ f
(3) ∥ ∥
∥∞ .
If {∆ k } is locally uniform with constant A, then the constants C r are independent on k
and
C r = 30(5/2)r+1 A 2 .
r!
Now we analyse the error E 1 at the points t i , i=1,...,k+3.
The logical scheme here proposed is similar to that one presented in [14] for the
error of spline derivative in the uniform case.
THEOREM 3. If f ∈ C 3 (I), for i=3,...,k+1, we obtain
(20) y ′ i − f′ i =− h2 i+1 H i+1+ h 2 i−1 H i−1− 2h 2 i H i
f (3)
i + O(h 3 i−1
48(h i−1 + h i )(h i + h i+1 )
),
where h i−1 is defined in (9) and
For i=1,2 we have
H i+1 = 2h i+1 (h i−1 + h i )+h i+2 (h i−1 + h i )−h i h i−1 ,
H i−1 = 2h i−1 (h i+1 + h i )+h i−2 (h i+1 + h i )−h i h i+1 ,
H i = 2h i (h i−1 + h i+1 )+h 2 i + 4h i+1h i−1 .
(21) y ′ 1 − f′ 1 =− h 2(2h 2 + h 3 )
24
where we define h 0 = max 0≤i≤3 h i and
f (3)
1
+ O(h 3 0 ),
(22) y ′ 2 − f′ 2 =− h2 3 (h 4+ 2h 3 )−2h 2 2 (h 2+ 2h 3 )
48(h 2 + h 3 )
and analogous results hold for i=k+2,k+3.
f (3)
2
+ O(h 3 1 ),