Para-orthogonal Laurent polynomials and the strong Stieltjes moment problem

On the space, A of Laurent polynomials we consider a linear functional L which is positive definite on (0,∞) and is defined in terms of a given bisequence, {ck}k=−∞∞. For each ω>0, we define a sequence {Nn(z,ω)}n=0∞ of rational functions in terms of two sequences of orthogonal Laurent polynomials, {Qn(z)}n=0∞ and {Q̂n(z)}n=0∞, which span A in the order {1,z−1,z,z−2,z2,…} and {1,z,z−1,z2,z−2,…}, respectively. It is shown that the numerators and denominators of each Nn(z,ω) are linear combinations of the canonical numerators and denominators of a modified PC-fraction. Consequently, {N2n(z,ω)}n=0∞ and {N2n+1(z,ω)}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.