Untitled - Cdm.unimo.it

36 Polynomial Approximation of Differential Equations
one considers the polynomial (x − ξ1)(x − ξ2)un and, if n > 2, deduces the existence
of a third zero, and so on. The procedure ends when n zeroes are finally obtained.
For any n ≥ 1 we denote the n zeroes of un by ξ (n)
k , 1 ≤ k ≤ n. We assume that these
are in increasing order (although many authors prefer the reverse order). It is obvious
that the polynomial u ′ n has n − 1 real distinct zeroes in I. We denote these zeroes by
η (n)
k , 1 ≤ k ≤ n − 1. One can prove that un and un−1 do not have common zeroes.
Moreover, between any two consecutive zeroes of un−1, there exists one and only one
zero of un.
Many general theorems characterize the zeroes of orthogonal polynomials. It suf-
fices to recall the following statement.
Theorem 3.1.2 - Let {un} n∈N be a sequence of orthogonal polynomials in I. Then,
for any interval [a,b] ⊂ I, a < b, it is possible to find m ∈ N such that um has at least
one zero in [a,b].
In other words, this theorem states that the set J =
n≥1
n k=1 {ξ(n)
k
} is dense in Ī.
We are going to review some properties of the zeroes of classical orthogonal poly-
nomials. We refer for instance to szegö(1939) for proofs and further results.
Zeroes in the Jacobi case - We restrict ourselves to the case where − 1
2
− 1
2
1 ≤ β ≤ 2 . Under these conditions, for n ≥ 1 we have the estimates
(3.1.1) −1 ≤ −cos
≤ α ≤ 1
2 and
k + (α + β − 1)/2
k
π ≤ ξ(n)
k ≤ −cos
π ≤ 1,
n + (α + β + 1)/2 n + (α + β + 1)/2
1 ≤ k ≤ n.
This shows that asymptotically the distance between two consecutive zeroes is propor-
tional to 1/n for points located in the central part of the interval I =] − 1,1[, and
proportional to 1/n 2 for points located near the endpoints of the same interval.