Orthogonality and recurrence for ordered Laurent polynomial sequences

We consider orderings of nested subspaces of the space of Laurent polynomials on the real line, more general than the balanced orderings associated with the ordered bases {1,z^-^1,z,z^-^2,z^2,...} and {1,z,z^-^1,z^2,z^-^2,...}. We show that with such orderings the sequence of orthonormal Laurent polynomials determined by a positive linear functional satisfies a three-term recurrence relation. Reciprocally, we show that with such orderings a sequence of Laurent polynomials which satisfies a recurrence relation of this form is orthonormal with respect to a certain positive functional.

[1]  T. Chihara,et al.  An Introduction to Orthogonal Polynomials , 1979 .

[2]  Carlos Díaz-Mendoza,et al.  Strong Stieltjes distributions and orthogonal Laurent polynomials with applications to quadratures and Padé approximation , 2005, Math. Comput..

[3]  Adhemar Bultheel,et al.  Quadrature on the half-line and two-point Pade´ approximants to Stieltjes functions—II: convergence , 1997 .

[4]  W. J. Thron,et al.  Continued Fractions: Analytic Theory and Applications , 1984 .

[5]  Haakon Waadeland,et al.  Continued fractions with applications , 1994 .

[6]  William B. Jones,et al.  Orthogonal Laurent polynomials and strong moment theory: a survey , 1999 .

[7]  Erik Hendriksen,et al.  Orthogonal Laurent polynomials , 1986 .

[8]  A. Bultheel,et al.  Quadrature on the half line and two-point Pade´ approximants to Stieltjes functions: subtitle: Part III. The unbounded case , 1997 .

[9]  A. Bultheel,et al.  Orthogonal rational functions on the real half line with poles in [-∞,0] , 2005 .

[10]  Adhemar Bultheel,et al.  Orthogonal Rational Functions , 1999, Cambridge monographs on applied and computational mathematics.

[11]  William B. Jones,et al.  Survey of continued fraction methods of solving moment problems and related topics , 1982 .

[12]  William B. Jones,et al.  A strong Stieltjes moment problem , 1980 .

[13]  P. González-Vera,et al.  Orthogonal Laurent polynomials and two-point Padé approximants associated with Dawson's integral , 2005 .

[14]  W. J. Thron,et al.  Encyclopedia of Mathematics and its Applications. , 1982 .

[15]  ORTHOGONAL LAURENT POLYNOMIALS AND QUADRATURES ON THE UNIT CIRCLE AND THE REAL HALF-LINE∗ , .

[16]  William B. Jones,et al.  Orthogonal Laurent polynomials and the strong Hamburger moment problem , 1984 .

[17]  Adhemar Bultheel,et al.  Quadrature on the half line and two-point Padé approximants to Stieltjes functions. Part I: A , 1995 .

[18]  William B. Jones,et al.  Orthogonal Laurent Polynomials and Gaussian Quadrature , 1981 .