Padé Table - The Block Theorem and Normal Approximants

The Block Theorem and Normal Approximants

Because of the way the (m, n)th approximant is constructed, the difference

Qn(z)f(z) − Pm(z)

is a power series whose first term is of degree no less than

m + n + 1.

If the first term of that difference is of degree

m + n + r + 1, r > 0,

then the rational function Rm, n occupies

(r + 1)2

cells in the Padé table, from position (m, n) through position (m+r, n+r), inclusive. In other words, if the same rational function appears more than once in the table, that rational function occupies a square block of cells within the table. This result is known as the block theorem.

If a particular rational function occurs exactly once in the Padé table, it is called a normal approximant to f(z). If every entry in the complete Padé table is normal, the table itself is said to be normal. Normal Padé approximants can be characterized using determinants of the coefficients cn in the Taylor series expansion of f(z), as follows. Define the (m, n)th determinant by

D_{m,n} = \left|\begin{matrix}
c_m & c_{m-1} & \ldots & c_{m-n+2} & c_{m-n+1}\\
c_{m+1} & c_m & \ldots & c_{m-n+3} & c_{m-n+2}\\
\vdots & \vdots & & \vdots & \vdots\\
c_{m+n-2} & c_{m+n-3} & \ldots & c_m & c_{m-1}\\
c_{m+n-1} & c_{m+n-2} & \ldots & c_{m+1} & c_m\\
\end{matrix}\right|

with Dm,0 = 1, Dm,1 = cm, and ck = 0 for k < 0. Then

  • the (m, n)th approximant to f(z) is normal if and only if none of the four determinants Dm,n−1, Dm,n, Dm+1,n, and Dm+1,n+1 vanish; and
  • the Padé table is normal if and only if none of the determinants Dm,n are equal to zero (note in particular that this means none of the coefficients ck in the series representation of f(z) can be zero).

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