Page 109 - Textos de Matemática Vol. 34

ORTHOGONAL POLYNOMIALS 99
Theorem 5.1. The operator F : P′ → ∆′ defined by u∈P′ →F(u):=ν≥0uνzν
is a topological isomorphism.
Proof. It is easy to see that F is linear and bijective. Now, notice that, for each u∈P′ andn∈N0
sn (F(u)) = sup |uν| =: |u|n . 0≤ν ≤n
Therefore, since {|u|n : n ∈ N0} defines the topology in P′ (by Theorem 2.1), we see that for each sequence (uk)k in P′ it holds
F(uk) → 0 (in ∆′) iff sn (F(uk)) → 0 for all n = 0,1,2,··· iff |uk|n → 0 for all n = 0,1,2,···
iff uk→0 (inP′).
It follows that F is bicontinuous.
The important fact in the present context is that the isomorphism F allows us to transpose the algebraic structure from ∆′ to P′.
Definition 5.1. Let u ∈ P′. The formal Stieltjes series associated with u is
Su(z):=−
zν+1 ≡−zF z
.
uν 1 1
ν≥0
This is an important concept and a very useful tool in order to characterize some classes of orthogonal polynomials (e.g., the so-called classical OP’s, semiclassical OP’s, or Laguerre-Hahn OP’s).
6. Sequences of orthogonal polynomials
We now introduce the formal definition of orthogonal polynomials. Our main
references are Szeg¨o [20], Chihara [4] and Freud [5].
Definition 6.1. Let u ∈ P∗. A sequence of polynomials (Pn)n is said to be an
orthogonal polynomial sequence (OPS) with respect to u if it is a simple set and ⟨u,PnPm⟩ = knδn,m , n,m = 0,1,2,··· ,
where kn is a nonzero complex number for each n. When there exists an OPS with respect to u, u is said to be regular (or quasi-definite).