Page 110 - Textos de Matemática Vol. 34

100 JOSE´ CARLOS PETRONILHO
It is easy to verify that if (Pn)n and (Qn)n are two OPS’s with respect to the same functional u ∈ P∗ then there exists a sequence of nonzero complex numbers (cn)n such that Qn = cnPn for all n = 0, 1, 2, · · · . Hence without loss of generality usually one works with monic orthogonal polynomial sequences (MOPS), which are OPS’s such that the leading coefficient of each polynomial is 1. Another useful normalization occurs when kn = 1 for all n. In such a case the OPS is called an orthonormal polynomial sequence.
One of the most important tools in the theory of orthogonal polynomials is the so called Favard’s theorem (see e.g. Chihara [4, p.21]), which states that any OPS {Pn}n≥0 is characterized by a three-term recurrence relation
(6.1) xPn(x) = αnPn+1(x) + βnPn(x) + γnPn−1(x) , n = 0, 1, 2, . . .
with initial conditions P−1(x) = 0 and P0(x) = const. ̸= 0, where {αn}n≥0, {βn}n≥0 and {γn}n≥0 are sequences of complex numbers such that αnγn+1 ̸= 0 for all n = 0,1,2,···. The monic case corresponds to αn = 1 for all n = 0,1,2,···. Under these conditions, the relations
βn = ⟨u,xPn2⟩ , γn+1 = ⟨u,Pn2+1⟩ ⟨u, Pn2⟩ ⟨u, Pn2⟩
hold for all n = 0,1,2,···. Further, the case when βn is a real number and γn+1 > 0 for all n = 0, 1, 2, · · · is of special importance in applications. In this case u can be represented by a Lebesgue-Stieltjes integral defined by some distribution function, i.e., a bounded and non-decreasing function σ : R → R, with an infinite set of increasing points2, such that P ⊂ L1(R,BR,μσ) and

(6.2) ⟨u,f⟩ =
f dμσ (f ∈ P),
R
μσ : BR → [0,+∞] being the Borel-Stieltjes measure defined by σ (i.e., μσ is the
unique Borel measure μ : BR → [0, +∞] such that μ([a, b)) = σ(b−) − σ(a+) for all a,b ∈ R with a < b) – cf. Chihara [4, Chapter 2]. The condition P ⊂ L1(R,BR,μσ) is equivalent to say that all the moments of μσ are finite, i.e.,
n
|x| dμσ < ∞, n = 0,1,2,··· .
R
We point out that an MOPS {Pn}n≥0 is orthogonal with respect to a unique mo- ment linear functional u ∈ P∗, but when u is represented by a Lebesgue-Stieltjes integral in the sense just mentioned above, then the function σ needs not to be unique. If, however, the support of μσ is a compact set then σ is unique. In par- ticular, when both sequences {βn}n≥0 and {γn}n≥1 in the three-term recurrence relation for the MOPS {Pn}n≥0 are bounded then the support of μσ is compact, and
A point ξ ∈ R is an increasing point of the function σ if σ(ξ + ε) − σ(ξ − ε) > 0 for all ε > 0.