ABSTRACT
P.n = (pn, 1)a = i: x"dp.a(x) and 1J.n = (pn, 1)H = /_: z"dpH(z). Now observe that each of the sequences {Gm(z)} and {Hm(z)}, defined in Lemma 16.1, is an example of the sequence {Sm{Y)} in Theorem 6.11. This latter sequence, on the other hand, can be identified with the sequence {Pn(z)} in [Ak1, p. 3), if the subscript of the Dn in [Ak1, p. 1) is increased by one. From [Ak1, p. 84, Prob. 10], we then find that the criteria for each of the measures J'G and PH to be respectively unique are those in {16.2), which proves the theorem. 0 Comments. Some specializations of Theorem 16.2 are:
1. The two respective measures given by (8.44), (8.45) and (8.47), {8.48) for the sequences {Am(z,k)} and {Bm{z,k)} are unique (up to normalization). {Here c1 = k2 + 1 and c2 = k2 .) We cannot conclude the same uniqueness result for the sequences {A:a(z,k)} and {B~(z,k)} in 8 (b), because the condition 2..fo2 $ let! in Theorem 16.2 does not hold. (Therec1 = 2{1-2k2 ) and c2 = 1, 0 < k < 1.)
2. The weight functions WA and ws, defined respectively in Definition 14.1 for the sequences {Am(z)}:=o and {Bm{z)}:=o• are "essentially unique" (in that other weight functions must be equal to these .\-a.e., .\ being Lebesgue measure). Corollary 16.3. Let f E :F2 be such that let! 2: 2..fo2. Also, form, n 2: 0, m 1= n, let p0 , P.H : l!(lR) --+ [0, 1) be the unique Borel probability measures for which JR Gm Gn dpa = 0 and JR Hm Hn dp.a = 0, respectively. Then 1' is norm-dense in each of the two real Hilbert spaces Lf{lll, l!(lll), JJG) and Lf(lll, !i(lll), JJH)· Proof. By Theorem 16.2, the measures JJG and JJH are unique. The density of 1' then follows from [Ak1, p. 45, Cor. 2.3.3). 0 Comment. As a result of Corollary 16.3, each of the four sequences {Am(z, k)}:=o• {Bm(z, k)}:=o (cf. Tables 8.3 and 8.4), and {Am{z)}:=O• {Bm(z)}:=O in (13.3) and (13.4) is an orthogonal basis ofthe corresponding ~space.
