Laurent Series and their Padé Approximations

2. Introduction.- 2.1 Classical Padé approximation.- 2.2 Toeplitz and Hankel systems.- 2.3 Continued fractions.- 2.4 Orthogonal polynomials.- 2.5 Rhombus algorithms and convergence.- 2.6 Block structure.- 2.7 Laurent-Padé approximants.- 2.8 The projection method.- 2.9 Applications.- 2.10 Outline.- 3. Moebius transforms, continued fractions and Padé approximants.- 3.1 Moebius transforms.- 3.2 Flow graphs.- 3.3 Continued fractions (CF).- 3.4 Formal series.- 3.5 Padé approximants.- 4. Two algorithms.- 4.1 Algorithm 1.- 4.2 Algorithm 2.- 5. All kinds of Padé Approximants.- 5.1 Padé approximants.- 5.2 Laurent-Padé approximants.- 5.3 Two-point Padé approximants.- 6. Continued fractions.- 6.1 General observations.- 6.2 Some special cases.- 7. Moebius transforms.- 7.1 General observations.- 7.2 Some special cases.- 8. Rhombus algorithms.- 8.1 The ab parameters (sawtooth path).- 8.2. The FG parameters (row path).- 8.3. A staircase path.- 8.4 ?? paramaters (diagonal path).- 8.5 Some dual results.- 8.6 Relation with classical algorithms.- 9. Biorthogonal polynomials, quadrature and reproducing kernels.- 9.1 Biorthogonal polynomials.- 9.2 Interpolatory quadrature methods.- 9.3 Reproducing kernels.- 9.4 Other orthogonality relations.- 10. Determinant expressions and matrix interpretations.- 10.1 Determinant expressions.- 10.2 Matrix interpretations.- 10.2.1 Toeplitz matrices.- 10.2.2 Hankel matrices.- 10.2.3 Tridiagonal matrices.- 11. Symmetry Properties.- 11.1 Symmetry for F(z) and $$\hat F$$(z) = F(1/z).- 11.2 Symmetry for F(z) and G(z) = 1/F(z).- 12. Block structures.- 12.1 Pade forms, Laurent-Pade forms and two-point Pade forms.- 12.2 The T-table.- 12.3 The Pade, Laurent-Pade, and two-point Pade tables.- 13. Meromorphic functions and asymptotic behaviour.- 13.1 The function F(z).- 13.2 Asymptotics for finite Toeplitz determinants.- 13.3 Asymptotics for infinite Toeplitz determinants.- 13.4 Consequences for the T-table.- 14. Montessus de Ballore theorem for Laurent-Padé approximants.- 14.1 Semi infinite Laurent series.- 14.2 Bi-infinite Laurent series.- 15. Determination of poles.- 15.1 Rutishauser polynomials of type 1 and type 2.- 15.2 Rutishauser polynomials of type 3.- 15.3 Rutishauser polynomials and Laurent series.- 15.4 Convergence of parameters.- 16. Determination of zeros.- 16.1 Dual Rutishauser polynomials and semi-infinite series.- 16.2 From semi-infinite to bi-infinite series.- 16.3 Convergence of parameters.- 17. Convergence in a row of the Laurent-Padé table.- 17.1 Toeplitz operators and the projection method.- 17.2 Convergence of the denominator.- 17.3 Convergence of the numerator.- 18. The positive definite case and applications.- 18.1 Function classes.- 18.2 Connection with the previous results.- 18.3 Stochastic processes and systems.- 18.4 Lossless inverse scattering and transmission lines.- 18.5 Laurent-Padé approximation and ARMA-filtering.- 18.6 Concluding remarks.- 19. Examples.- 19.1 Example 1.- 19.2 Example 2.- 19.3 Example 3.- References.- List of symbols.