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Block structures

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

Abstract

Until now we supposed that the fls F(z) was normal which implies that
$$S_n^{(m)}(z)$$
and
$$\hat S_n^{(m)}(z)$$
where S is one of A, Q, P, B(p), R existed for all m = 0, ±1,±2,… and n = 0,1,2,…. All the PAs
$$P_n^{(m)}(z)/Q_n^{(m)}(z)$$
and
$$\hat P_n^{(m)}(z)/\hat Q_n^{(m)}(z)$$
, all the two-point PAs
$$A_n^{(m)}(z)/Q_n^{(m)}(z)$$
and
$$\hat A_n^{(m)}(z)/\hat Q_n^{(m)}(z)$$
and all LPAs
$$K_n^{(m)}(z)$$
and
$$\hat K_n^{(m)}(z)$$
existed and were unique for each value of m and n. We shall investigate what happens if F(z) is not normal, i.e. if the Toeplitz determinants
$$T_n^{(m)}$$
are not all nonzero. It is well known that for a fps F(z), the table with (m,n)th entry
$$T_n^{(m)}$$
has a so called block structure. This means that zero entries in the T-table show up in square blocks. The block structure of the T-table also introduces a block structure in the table of Padé approximants, of Laurent-Padé approximants and two-point Padé approximants. In fact, if the fls is not normal, some of these approximants do not exist. However, Padé forms, Laurent-Padé forms and two-point Padé forms will always exist according to our definitions of section 2.5. In chapter 4 we have seen how these approximants could be obtained from the algorithms of chapter 3. More precisely, we have found these approximants in terms of the quantities
$$Q_n^{(m)}(z)$$
,
$$A_n^{(m)}(z)$$
,
$$B(p)_n^{(m)}(z)$$
that appeared in the algorithms of chapter 3. However they were defined under the conditions that F(z) were a normal fls. Since now F(z) is not normal any more, we shall redefine these polynomials and also the series
$$P_n^{(m)}(z)$$
and
$$R_n^{(m)}(z)$$
in a weaker form.

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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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