Page 111 - Textos de Matemática Vol. 34

ORTHOGONAL POLYNOMIALS 101
so the uniqueness of σ holds. Further, when σ is unique then the set of increasing points of σ coincides with the support of the measure μσ.
7. Orthogonal polynomials and dual basis
Every OPS is a simple set of polynomials, hence it has an associated dual basis in P′. The next proposition establishes some connections between a regular moment linear functional in P and the dual basis associated to the corresponding OPS.
Theorem 7.1. Let u ∈ P∗ be regular, (Pn)n the corresponding MOPS, and (an)n the associated dual basis. Then the following statements hold:
(i) The functionals in the dual basis are explicitly given by an=Pn(x)u, n=0,1,2,···.
n ≥ N + 1, then
N
v = ⟨v,Pν⟩aν .
ν=0
As a consequence, there exists φ ∈ P, with ∂φ ≤ N, such that
v=φu.
This polynomial φ is given by φ(x) = N ⟨v, Pν ⟩ Pν (x) . Further, ∂φ = N
ν=0 ⟨u,Pν2⟩
if and only if ⟨v,PN⟩ ≠ 0.
(iii) u = u0 a0, so that {Pn}n≥0 is orthogonal with respect to a0.
(iv) If the three-term recurrence relation satisfied by the MOPS (Pn)n is (6.1)
(with αn = 1 for all n) then
xan =an−1+βnan+γn+1an+1 , n=1,2,···.
⟨ u , P n2 ⟩
(ii) If v ∈ P∗ and there exists an integer N ∈ N0 such that ⟨v,Pn⟩ = 0 for all
Proof. For every n,ν = 0,1,2,···, we have
⟨ Pn u,Pν ⟩= 1 ⟨u,PnPν ⟩=δn,ν =⟨an,Pν⟩,
⟨u, Pn2⟩ ⟨u, Pn2⟩
so the action of the linear functionals Pn u and a over each element of the
⟨u,Pn2 ⟩ n
basis (Pν)ν of P coincides. Thus the action of these functionals coincides over all
the space P. This proves (i). Statement (ii) is an straightforward consequence of Theorem 3.1 and (i), and (iii) follows from (ii) taking v = u (which implies N = 0).