A note on recursively defined orthogonal polynomials
1969; Mathematical Sciences Publishers; Volume: 28; Issue: 3 Linguagem: Inglês
10.2140/pjm.1969.28.611
ISSN1945-5844
Autores Tópico(s)Matrix Theory and Algorithms
ResumoLet {αJΓ-o and {δt}Π= 0 ^e rea l sequences and suppose the bi,β are all positive.Define a sequence of polynomials {P;(sc)}Π=o as follows: P 0 (cc) = 1, Pi(x) = (x -ao)lb o , and for n ^ 1 (*) bnPn+i(x) = (a -α»)iVα;) -&«-iP»-i(αO .Favard showed that the polynomials {P τ (x)} are orthonormal with respect to a bounded increasing function γ defined on (-co, +oo).This note generalizes recent constructive results which deal with connections between the two sequences {a τ } and {bi} and the spectrum of f.(The spectrum of ψ is the set S(ψ) = {λ : f(λ + e)-f(λ -e)>0 for all ε > 0}.)It is shown that if bi -> 0 then every limit point of the sequence {α^} is in S(f). Preliminaries* In order to use theorems from functionalS +oo f 2 dψ < co}.This is a -co jHubert space where the inner product is gived by (/, g) = I fgdψ and where we identify all functions which agree on S(ψ).In [2], (p.215), Carleman showed that the condition X 1/V b { = oo implies that when ψ is normalized to be continuous from the left and to have ijr(-co) = 0, ψ( + ) -1, then it is unique.In [6], M. Riesz showed that if ψ is essentially unique then ParsevaFs relation holds for the orthonormal set {P { } in the space ^f2 {ψ).Hence the set {PJ is dense in this space.We now make the assumption that lim 6 { = 0. Combining the Carleman result and the Riesz result we see that ψ is essentially unique and the polynomials {P<} are a dense set in ^f\ψ).Using this information we define an operator A on a dense subset of ^f\ψ).The domain of A is the set of all functions / which are in ^2(ψ) and for which xf is also in J*?\ψ).We take A to be the self-ad joint operator defined by (Af)(x) = xf(x).By inspection of (*) we see that for i = 1, 2, 3, we haveWe call A the operator associated with the sequences {α<} and {bi}.3* Theorems* Let σ(A) be the spectrum of the operator A, i.e., all points λ where A -λl does not have a bounded inverse.Then we have the following:
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