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Transforms 67
It is evident that K −1
n = D (1)
n K t nD (2)
n , where D (1) and D (2) are the diagonal matrices
given respectively by diag{uj 2 w}0≤j≤n−1 and diag{1/w (n)
i+1 }0≤i≤n−1. This shows
that, after an appropriate normalization of the elements of the two bases, the inverse of
Kn is given by its transpose.
Similar arguments can be applied to the analysis of transforms based on Gauss-
Lobatto nodes. Here the space Pn is generated either by {uk}0≤k≤n or by { ˜ l (n)
j }0≤j≤n.
Hence, we introduce a discrete Fourier transform ˜ Kn : Pn → Pn, which corresponds
to an (n + 1) × (n + 1) matrix { ˜ kij} 0≤i≤n . To get the first n − 1 coefficients we use
0≤j≤n
(2.3.7) and (3.5.1):
(4.1.8) ci =
1
ui 2 w
n
j=0
For the last coefficient we recall (3.8.12)
(4.1.9) cn =
p(η (n)
j ) ui(η (n)
j ) ˜w (n)
j , 0 ≤ i ≤ n − 1.
1
un 2 w,n
Conversely, thanks to (2.3.1), one has
(4.1.10) p(η (n)
i ) =
n
j=0
n
j=0
p(η (n)
j ) un(η (n)
j ) ˜w (n)
j .
cj uj(η (n)
i ), 0 ≤ i ≤ n.
Finally, for the Laguerre Gauss-Radau case, we define ˜ Kn : Pn−1 → Pn−1 as described
above. Then, we have by virtue of (2.3.7) and (3.6.1)
(4.1.11) ci =
1
L (α)
i 2 w
(4.1.12) p(η (n)
i ) =
n−1
j=0
n−1
j=0
p(η (n)
j ) L (α)
i (η (n)
j ) ˜w (n)
j , 0 ≤ i ≤ n − 1,
cj L (α)
j (η (n)
i ), 0 ≤ i ≤ n − 1.
The scaled weights introduced in section 3.10 can be taken into account in the evaluation
of (4.1.11).