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Convergence in a row of the Laurent-Padé table

  • Adhemar Bultheel
Part of the Operator Theory: Advances and Applications book series (OT, volume 27)

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
$$\mathop {\lim }\limits_{n \to \infty } Q_n^{(m)}(z)$$
first. This problem, or a variant thereof, has been studied in the literature. The vector
$$Q_n^{(m)}$$
of coefficients of the polynomial
$$Q_n^{(m)}(z)$$
is the solution of a (n+1) × (n+1) Toeplitz system of linear equations. If n → ∞, the matrix
$${\rm T}_n^{(m)}$$
of the system will become an infinite Toeplitz matrix. It is the representation in the natural basis
$$\{ {z^k}\} _{ - \infty }^\infty $$
of a Toeplitz operator with symbol 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
with
$${\rm T}_n^{(m)} = ({f_{m + i - j}})_{i,j}^n = 0\,\,\,\, \in \,\,\,\,\,{\mathbb{C}^{(n + 1) \times (n + 1)}}$$
,
$${E_{0,n}} = {[1\,\,0 \cdot \cdot \cdot 0]^T}\,\,\,\, \in \,\,\,{\mathbb{C}^{n + 1}}$$
and
$${Q_n}\,\, \in \,\,{\mathbb{C}^{n + 1}}$$
will become
$${\rm T}_n^{(m)}{Q_\infty } = {E_{0,\infty }}$$
(16.2)
which is the representation in the natural basis of a Toeplitz operator equation
$$\Phi Q(z) = 1$$
(16.3)
where Φ represents the Toeplitz operator with special symbol 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
$$\{ {f_k}\} _{ - \infty }^\infty $$
, were the Fourier coefficients of F(z) ∈ L1 and for Q(z) ∈Hp. 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
$$Q_n^{(0)}(z)$$
uniformly on compacts if the winding number κ of F(z) is zero.

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Copyright information

© Birkhäuser Verlag Basel 1987

Authors and Affiliations

  • Adhemar Bultheel
    • 1
  1. 1.Dept. Computer ScienceK. U. LeuvenLeuven-HeverleeBelgium

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