Author | Bultheel, Adhemar. author |
---|---|
Title | Laurent Series and their Padรฉ Approximations [electronic resource] / by Adhemar Bultheel |
Imprint | Basel : Birkhรคuser Basel, 1987 |
Connect to | http://dx.doi.org/10.1007/978-3-0348-9306-0 |
Descript | XI, 276 p. online resource |
2. Introduction -- 2.1 Classical Padรฉ 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-Padรฉ approximants -- 2.8 The projection method -- 2.9 Applications -- 2.10 Outline -- 3. Moebius transforms, continued fractions and Padรฉ approximants -- 3.1 Moebius transforms -- 3.2 Flow graphs -- 3.3 Continued fractions (CF) -- 3.4 Formal series -- 3.5 Padรฉ approximants -- 4. Two algorithms -- 4.1 Algorithm 1 -- 4.2 Algorithm 2 -- 5. All kinds of Padรฉ Approximants -- 5.1 Padรฉ approximants -- 5.2 Laurent-Padรฉ approximants -- 5.3 Two-point Padรฉ 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 -- 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-Padรฉ 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-Padรฉ 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-Padรฉ 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