2Orthogonal polynomials
IB Numerical Analysis
2.4 Examples
The four famous examples are the Legendre polynomials, Chebyshev polynomials,
Laguerre polynomials and Hermite polynomials. We first look at how the
Chebyshev polynomials fit into this framework.
Chebyshev is based on the scalar product defined by
hf, gi =
Z
1
−1
1
√
1 −x
2
f(x)g(x) dx.
Note that the weight function blows up mildly at the end, but this is fine since
it is still integrable.
This links up with
T
n
(x) = cos(nθ)
for x = cos θ via the usual trigonometric substitution. We have
hT
n
, T
m
i =
Z
π
0
1
√
1 −cos
2
θ
cos(nθ) cos(mθ) sin θ dθ
=
Z
π
0
cos(nθ) cos(mθ) dθ
= 0 if m 6= n.
The other orthogonal polynomials come from scalar products of the form
hf, gi =
Z
b
a
w(x)f(x)g(x) dx,
as described in the table below:
Type Range Weight
Legendre [−1, 1] 1
Chebyshev [−1, 1]
1
√
1−x
2
Laguerre [0, ∞) e
−x
Hermite (−∞, ∞) e
−x
2