Laurent Series and their Padé Approximations pp 141-154 | Cite as

# Block structures

Chapter

## Abstract

Until now we supposed that the fls and where and , all the two-point PAs and and all LPAs and existed and were unique for each value of are not all nonzero. It is well known that for a fps has a so called block structure. This means that zero entries in the , , that appeared in the algorithms of chapter 3. However they were defined under the conditions that and in a weaker form.

*F*(*z*) was normal which implies that$$S_n^{(m)}(z)$$

$$\hat S_n^{(m)}(z)$$

*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$$P_n^{(m)}(z)/Q_n^{(m)}(z)$$

$$\hat P_n^{(m)}(z)/\hat Q_n^{(m)}(z)$$

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

$$\hat A_n^{(m)}(z)/\hat Q_n^{(m)}(z)$$

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

$$\hat K_n^{(m)}(z)$$

*m*and*n*. We shall investigate what happens if*F*(*z*) is not normal, i.e. if the Toeplitz determinants$$T_n^{(m)}$$

*F*(*z*), the table with (*m,n*)th entry$$T_n^{(m)}$$

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

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

$$B(p)_n^{(m)}(z)$$

*F*(*z*) were a normal fls. Since now*F*(*z*) is not normal any more, we shall redefine these polynomials and also the series$$P_n^{(m)}(z)$$

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

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## Copyright information

© Birkhäuser Verlag Basel 1987