Page 112 - Textos de Matemática Vol. 34

102 JOSE´ CARLOS PETRONILHO
To prove (iv) notice first that, again by Theorem 3.1, xan = ν≥0⟨xan,Pν⟩aν
for all n. But, for all n = 1,2,··· and ν = 0,1,2,···, we have ⟨xan,Pν⟩ = ⟨an,xPν⟩ = ⟨an,Pν+1 +βνPν +γνPν−1⟩
= ⟨an−1+βnan+γn+1an+1,Pν⟩,
Corollary 7.2. Let (Pn)n be a given MOPS (with respect to some functional in
P∗) and let v ∈ P∗. Then (Pn)n is a MOPS with respect to v if and only if
(7.1) ⟨v,1⟩ ≠ 0, ⟨v,Pn⟩ = 0, n = 1,2,3,··· .
Proof. If (Pn)n is a MOPS with respect to v then it is clear that (7.1) holds. Conversely, if (Pn)n is a MOPS such that (7.1) holds then by (ii) in Theorem 3.1 v = ⟨v,1⟩a0, where a0 is the first element in the dual basis associated to (Pn)n. Therefore, since by (iii) in Theorem 3.1 (Pn)n is orthogonal with respect to a0 and, by hypothesis, ⟨v, 1⟩ ≠ 0, it follows that (Pn)n is a MOPS with respect to v.
8. An inverse problem: linearly related OPS’s
The results in this section are motivated by a recent paper by M. Alfaro, F. Marcell´an, A. Pen˜a and M. L. Rezola [1], where the authors characterized linearly related sequences of orthogonal polynomials in terms of their functionals. In fact, one of their main results (cf. [1, Theorem 2.4]) asserts that if (Pn)n and (Qn)n are two MOPS’s such that the structure relation
(8.1) Pn(x) + rnPn−1(x) = Qn(x) + snQn−1(x)
holds for all n = 0, 1, 2, · · · (with some additional extra –but natural– conditions)
then the corresponding moment linear functionals u and v (resp.) satisfy
(8.2) (x − a)u = λ(x − b)v ,
where a, b and λ are complex numbers (with λ ̸= 0). Conversely, they showed that a relation such as (8.2) between regular functionals u and v implies a structure relation as (8.1) for the corresponding MOPS’s. Special cases of these type of relations were treated in [7], [9] and [14]. The analysis of this kind of problems is motivated by the study of the so-called Sobolev-type orthogonal polynomials, namely sequences of polynomials which are orthogonal with respect to nonstandard scalar inner products
ν=0 R
supports are infinite sets, and with finite moments, i.e., |x|sdμν < ∞ for all R
which gives (iv).

N ( ν ) ( ν )
f g dμν,
(f,g):=
where μ0, · · · , μN are positive Borel measures supported on the real line, their