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70 Polynomial Approximation of Differential Equations
Thus, it is natural to define, for n ≥ 1, the operator Aw,n = Iw,n − Πw,n−1. The
polynomial in Pn−1
(4.2.1) Aw,nf = Iw,nf − Πw,n−1f, n ≥ 1, f ∈ C 0 ([−1,1]),
takes the name of aliasing error.
Depending on the smoothness of f, this error tends to zero when n → +∞. The proof
of this fact and the characterization of the aliasing error is given in section 6.6.
Another type of aliasing error is obtained by setting
(4.2.2) Ãw,nf = Ĩw,nf − Πw,nf, f ∈ C 0 ([−1,1]),
where n ≥ 1. This is also studied later on.
4.3 Fast Fourier transform
Let us examine what happens by applying the discrete Fourier transform in the special
case of the Chebyshev basis. By virtue of (1.5.6), (3.1.4), (3.4.6) and (2.2.12), the entries
of Kn in (4.1.5) take the form
(4.3.1) kij =
(−1) i
n
1 if i = 0, 0 ≤ j ≤ n − 1
i(2j + 1)π
cos ×
2n
2 if 1 ≤ i ≤ n − 1, 0 ≤ j ≤ n − 1
For the same reason, due to (3.1.11), (3.5.7) and (3.8.5), the entries of ˜ Kn are
(4.3.2) ˜ kij =
(−1) i
n
ijπ
cos ×
n
⎧
1
2 if i = 0, j = 0 or j = n
⎪⎨
1
1
if i = 0, 1 ≤ j ≤ n − 1
if 1 ≤ i ≤ n − 1, j = 0 or j = n
2
1 ⎪⎩ 2
1
if 1 ≤ i ≤ n − 1, 1 ≤ j ≤ n − 1
if i = n, j = 0 or j = n
if i = n, 1 ≤ j ≤ n − 1
.
.