Untitled - Cdm.unimo.it

Transforms 73
After evaluating the γm’s by (4.3.3), the output data can be recovered by
(4.3.7)
cm = (−1)m
2n ·
⎧
⎪⎨
⎪⎩
p(η (n)
0 ) + p(η (n)
n ) + 2 n−1
j=1 p(η(n)
j ) m = 0,
γm
1 + 2sin πm
n
−1
+ γn−m 1 − 2sin πm
−1
n
p(η (n)
0 ) + p(η (n)
n ) + 2 n−1
j=1 (−1)j p(η (n)
j ) m = n.
1 ≤ m ≤ n − 1,
Here the upper bar denotes complex conjugate. The cumbersome verification of (4.3.7)
is left to the reader. We note that the output coefficients in (4.3.7) are real. Other sug-
gestions and improvements are contained for instance in swarztrauber(1986). Again,
once the γm’s have been determined, the cost to implement (4.3.7) is proportional to
n.
We proceed now with the analysis of relation (4.3.3) and we consider the special
case when n = 8 = 2 3 . The matrix corresponding to the linear transformation takes the
form
(4.3.8) Φ =
⎡
⎢
⎣
φ0 φ0 φ0 φ0 φ0 φ0 φ0 φ0 φ0 φ1 φ2 φ3 φ4 φ5 φ6 φ7 φ0 φ2 φ4 φ6 φ0 φ2 φ4 φ6 φ0 φ3 φ6 φ1 φ4 φ7 φ2 φ5 φ0 φ4 φ0 φ4 φ0 φ4 φ0 φ4 φ0 φ5 φ2 φ7 φ4 φ1 φ6 φ3 φ0 φ6 φ4 φ2 φ0 φ6 φ4 φ2 φ0 φ7 φ6 φ5 φ4 φ3 φ2 φ1 where φ := e iπ/4 and we noted that φ k+8 = φ k , ∀k ∈ N.
Moreover, the linear system can be also written as
⎡ ⎤
⎡
(4.3.9)
⎢
⎣
γ0
γ4
γ2
γ6
γ1
γ5
γ3
γ7
⎥
⎦
= Φ1 Φ2 Φ3
where the matrices Φk, 1 ≤ k ≤ 3, are given by
⎢
⎣
δ0
δ1
δ2
δ3
δ4
δ5
δ6
δ7
⎤
⎥
⎥,
⎥
⎦
⎤
⎥
⎥,
⎥
⎦