Para-orthogonal Laurent polynomials and the strong Stieltjes moment problem

✍ Scribed by Catherine M. Bonan-Hamada; William B. Jones

• Publisher: Elsevier Science
• Year: 1999
• Tongue: English
• Weight: 113 KB
• Volume: 105
• Category: Article
• ISSN: 0377-0427
• DOI: 10.1016/s0377-0427(99)00024-2

✦ Synopsis

On the space, A of Laurent polynomials we consider a linear functional L which is positive deÿnite on (0; ∞) and is deÿned in terms of a given bisequence, {c k } ∞ k=-∞ . For each ! ¿ 0, we deÿne a sequence {Nn(z; !)} ∞ n=0 of rational functions in terms of two sequences of orthogonal Laurent polynomials, {Qn(z)} ∞ n=0 and { Qn (z)} ∞ n=0 , which span A in the order {1; z -1 ; z; z -2 ; z 2 ; : : :} and {1; z; z -1 ; z 2 ; z -2 ; : : :}, respectively. It is shown that the numerators and denominators of each Nn(z; !) are linear combinations of the canonical numerators and denominators of a modiÿed PCfraction. Consequently, {N2n(z; !)} ∞ n=0 and {N2n+1(z; !)} ∞ n=0 converge uniformly on compact subsets of C-{0} to analytic functions and hence lead to additional solutions to the strong Stieltjes moment problem.