Let {ai}i=0∞ and {bi}i=0∞ be
real sequences and suppose the bi’s are all positive. Define a sequence of
polynomials {Pi(x)}i=0∞ as follows: P0(x) = 1, P1(x) = (x − a0)∕b0, and for
n ≧ 1
(*)
Favard showed that the polynomials {Pi(x)} are orthonormal with respect to a
bounded increasing function ψ defined on (−∞,+∞). This note generalizes
recent constructive results which deal with connections between the two
sequences {ai} and {bi} and the spectrum of ψ. (The spectrum of ψ is the set
S(ψ) = {λ : ψ(λ + 𝜖) − ψ(λ − 𝜖) > 0 for all 𝜖 > 0}.) It is shown that if bi→ 0 then
every limit point of the sequence {ai} is in S(ψ).
