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