Page 334 - Elements of Distribution Theory

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320 Orthogonal Polynomials

10.6 Let {p 0 , p 1 ,...} denote orthogonal polynomials with respect to a distribution function F and
suppose that the polynomials are standardized so that
∞

2
p j (x) dF(x) = 1, j = 0, 1,....
−∞
Fix n = 0, 1,... and let K n denote the function defined in Exercise 10.5.
Show that for any polynomial g of degree n or less and any z ∈ R,
∞

2
2
g(z) ≤ K n (z, z) g(x) dF(x).
−∞
10.7 Let {p 0 , p 1 ,...} denote orthogonal polynomials with respect to a distribution function F and
n
suppose that the polynomials are standardized so that the coefficient of x in p n (x)is1.Let
n
G n denote the set of all polynomials of degree n with coefficient of x equal to 1. Find the
function g ∈ G n that minimizes

∞
2
g(x) dF(x).
−∞
10.8 Let {p 0 , p 1 ,...} denote orthogonal polynomials with respect to a distribution function F and
n
suppose that the polynomials are standardized so that the coefficient of x in p n (x)is1.For
each n = 1, 2,... let β n denote the coefficient of x n−1 in p n (x) and let

∞
2
h n = p n (x) dF(x).
−∞
Show that p 0 , p 1 ,... satisfy the three-term recurrence relationship
h n
p n+1 (x) = (x + β n+1 − β n )p n (x) − p n−1 (x).
h n−1
10.9 Let {p 0 , p 1 ,...} denote orthogonal polynomials with respect to a distribution function F. Show
that, for all m, n = 0, 1,...,
n+m

p m (x)p n (x) = a( j, m, n)p j (x)
j=0
where the constants a(0, m, n), a(1, m, n),... are given by

∞

p m (x)p n (x)p j (x) dF(x)
−∞
a( j, m, n) = .
∞ 2
p j (x) dF(x)
−∞
10.10 Prove Corollary 10.1.
10.11 Prove Corollary 10.2.
10.12 Prove Corollary 10.3.
10.13 Find the three-term recurrence relationship (see Exercise 10.8) for the Hermite polynomials.
10.14 Some authors define the Hermite polynomials to be polynomials orthogonal with respect to
the absolutely continuous distribution with density function

1
c exp − x 2 , −∞ < x < ∞,
2
where c is a constant. Let ¯ H 0 , ¯ H 1 ,... denote orthogonal polynomials with respect to this
n
n
distribution, standardized so that the coefficient of x in ¯ H n (x)is2 . Show that
n √
¯ H n (x) = 2 2 H n (x 2), n = 0, 1,....