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24 Polynomial Approximation of Differential Equations
In particular, theorem 2.2.1 shows that all the families of polynomials introduced in
chapter one are orthogonal with respect to the inner product (2.1.8), where w is their
corresponding weight function. More precisely, for any n,m ∈ N, n = m, one has
(2.2.3) (Jacobi)
(2.2.4) (Legendre)
(2.2.5) (Chebyshev)
(2.2.6) (Laguerre)
(2.2.7) (Hermite)
1
−1
P (α,β)
n (x)P (α,β)
m (x) (1 − x) α (1 + x) β dx = 0,
+∞
0
1
1
−1
+∞
−∞
−1
Pn(x)Pm(x) dx = 0,
Tn(x)Tm(x)
dx
√ 1 − x 2
= 0,
L (α)
n (x)L (α)
m (x) x α e −x dx = 0,
Hn(x)Hm(x) e −x2
dx = 0.
Note that scaled Laguerre (or Hermite) functions are not orthogonal.
Moreover, considering that b ≡ 0, (2.2.1) and (2.2.2) imply that the orthogonal polyno-
mials also satisfy
(2.2.8)
I
au ′ nu ′ m dx = 0, ∀n,m ∈ N with n = m.
This shows that the derivatives are orthogonal polynomials with respect to the weight
function a.
For any n ≥ 1, if p is a polynomial of degree at most n − 1 we have
since p is a linear combination of the uk’s for k ≤ n − 1.
I punw dx = 0,
By taking x = cos θ in (2.2.5) and recalling (1.5.6), one obtains the well-known
orthogonality relation for trigonometric functions (see for instance zygmund(1988)):
(2.2.9)
π
0
cos nθ cos mθ dθ = 0, ∀n,m ∈ N, n = m.
This will allows us further characterize Chebyshev polynomials.