Laurent Series and their Padé Approximations

The Pade approximation problem is, roughly speaking, the local approximation of analytic or meromorphic functions by rational ones. It is known to be important to solve a large scale of problems in numerical analysis, linear system theory, stochastics and other fields. There exists a vast literature on the classical Pade problem. However, these papers mostly treat the problem for functions analytic at 0 or, in a purely algebraic sense, they treat the approximation of formal power series. For certain problems however, the Pade approximation problem for formal Laurent series, rather than for formal power series seems to be a more natural basis. In this monograph, the problem of Laurent-Pade approximation is central. In this problem a ratio of two Laurent polynomials in sought which approximates the two directions of the Laurent series simultaneously. As a side result the two-point Pade approximation problem can be solved. In that case, two series are approximated, one is a power series in z and the other is a power series in z-l. So we can approximate two, not necessarily different functions one at zero and the other at infinity.

Contenu

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.