Laurent Series and their Padé Approximations pp 195-205 | Cite as

# Convergence in a row of the Laurent-Padé table

Chapter

## Abstract

In chapter 13 we have studied the Montessus de Ballore theorem for Laurent-Padé approximants. It gave the convergence of LPAs when the numerator degree tends to infinity. In this chapter we shall study the convergence of LPAs when the denominator degree tends to infinity. The idea is to find first. This problem, or a variant thereof, has been studied in the literature. The vector of coefficients of the polynomial is the solution of a ( of the system will become an infinite Toeplitz matrix. It is the representation in the natural basis of a Toeplitz operator with symbol with , and will become which is the representation in the natural basis of a Toeplitz operator equation where Φ represents the Toeplitz operator with special symbol , were the Fourier coefficients of uniformly on compacts if the winding number

$$\mathop {\lim }\limits_{n \to \infty } Q_n^{(m)}(z)$$

$$Q_n^{(m)}$$

$$Q_n^{(m)}(z)$$

*n*+1) × (*n*+1) Toeplitz system of linear equations. If*n →*∞, the matrix$${\rm T}_n^{(m)}$$

$$\{ {z^k}\} _{ - \infty }^\infty $$

*z*^{ -m }*F*(*z*). Thus for*n =*∞, the finite Toeplitz systems$${\rm T}_n^{(m)}{Q_n} = {E_{0,n}}\,\,\,\,\,\,\,\,,\,\,\,\,\,\,\,\,n = 0,1,2, \cdot \cdot \cdot $$

(16.1

$${\rm T}_n^{(m)} = ({f_{m + i - j}})_{i,j}^n = 0\,\,\,\, \in \,\,\,\,\,{\mathbb{C}^{(n + 1) \times (n + 1)}}$$

$${E_{0,n}} = {[1\,\,0 \cdot \cdot \cdot 0]^T}\,\,\,\, \in \,\,\,{\mathbb{C}^{n + 1}}$$

$${Q_n}\,\, \in \,\,{\mathbb{C}^{n + 1}}$$

$${\rm T}_n^{(m)}{Q_\infty } = {E_{0,\infty }}$$

(16.2)

$$\Phi Q(z) = 1$$

(16.3)

*z*^{ -m }*F*(*z*). The method of solving the operator equation (16.3) by successively solving the systems (16.1) for*n*= 0, 1, 2, … is called the projection method. This method and its convergence was studied by Gohberg and Feldman [GOF]. These authors studied the problem when$$\{ {f_k}\} _{ - \infty }^\infty $$

*F*(*z*) ∈ L^{1}and for*Q*(*z*) ∈H^{p}. Their result was that there can only be convergence of*Q*_{ n }to*Q*_{∞}(the Fourier coefficients of*Q*(*z*)) in*L*^{ p }-norm for one specific value of*m*. We shall slightly adapt their development in section 16.1 and we shall use it to prove convergence of$$Q_n^{(0)}(z)$$

*κ*of*F*(*z*) is zero.## Preview

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## Copyright information

© Birkhäuser Verlag Basel 1987