Theory of Statistics - George Mason University

0.1 Measure, Integration, and Functional Analysis 745
Table 0.2. Orthogonal Polynomials
Polynomial Weight
Series Range Function
Legendre [−1, 1] 1 (uniform)
Chebyshev [−1, 1] (1 − x 2 ) 1/2 (symmetric beta)
Jacobi [−1, 1] (1 − x) α (1 + x) β (beta)
Laguerre [0, ∞[ x α−1 e −x (gamma)
Hermite ] − ∞, ∞[ e −x2 /2 (normal)
Orthogonal polynomials are often expressed in the simpler, unnormalized
form. The first few unnormalized Legendre polynomials are
P0(t) = 1 P1(t) = t
P2(t) = (3t 2 − 1)/2 P3(t) = (5t 3 − 3t)/2
P4(t) = (35t 4 − 30t 2 + 3)/8 P5(t) = (63t 5 − 70t 3 + 15t)/8
(0.1.93)
The normalizing constant that relates the k th unnormalized Legendre polynomial
to the normalized form is determined by noting
1
−1
(Pk(t)) 2 dt = 2
2k + 1 .
The recurrence formula for the Legendre polynomials is
Pk(t) =
2k − 1
tPk−1(t) −
k
k − 1
k Pk−2(t). (0.1.94)
The Hermite polynomials are orthogonal with respect to a Gaussian, or
standard normal, weight function. We can form the normalized Hermite polynomials
using the Gram-Schmidt transformations on 1, x, x 2 , . . ., with a weight
function of e x/2 similarly to what is done in equations (0.1.92).
The first few unnormalized Hermite polynomials are
H e 0(t) = 1 H e 1(t) = t
H e 2(t) = t 2 − 1 H e 3(t) = t 3 − 3t
H e 4(t) = t 4 − 6t 2 + 3 H e 5(t) = t 5 − 10t 3 + 15t
(0.1.95)
These are not the standard Hermite polynomials, but they are the ones most
commonly used by statisticians because the weight function is proportional
to the normal density.
The recurrence formula for the Hermite polynomials is
H e k(t) = tH e k−1(t) − (k − 1)H e k−2(t). (0.1.96)
Theory of Statistics c○2000–2013 James E. Gentle