Block structures

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


Until now we supposed that the fls F(z) was normal which implies that
$$\hat S_n^{(m)}(z)$$
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
$$\hat P_n^{(m)}(z)/\hat Q_n^{(m)}(z)$$
, all the two-point PAs
$$\hat A_n^{(m)}(z)/\hat Q_n^{(m)}(z)$$
and all LPAs
$$\hat K_n^{(m)}(z)$$
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
are not all nonzero. It is well known that for a fps F(z), the table with (m,n)th entry
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
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
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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