Recurrence relations in Padé-Hermite approximation
Padé-Hermite approximation for a normal system of \(s>1\) power series is discussed. Then Padé-Hermite form (PHF) \(\phi\) (L) of each order \(L=(\ell_ 1,\ell_ 2,...,\ell_ s)\) exists and is defined uniquely up to a constant factor. Recurrence formulas are given which permit movement in the s-dimensional table of all PHFs in two different ways: (1) across this table (the sum \(\ell_ 1+\ell_ 2+...+\ell_ s\) assumes only two neighbouring values), (2) along this table (a sequence \(\{\phi (L_ j)\}\) is calculated, where \(L_{-1}:=(-1,-1,...,-1)\), \(L_ j:=L_{j- 1}+E_{d_ j}\), \(j=0,1,...\); \(E_ d\) is the dth unit vector). All earlier formulas of the second kind generalize Padé's idea and are of the order \(s: \phi (L_ n)=\sum^{n-1}_{j=n^*}(\beta_ jx+\gamma_ j)\phi (L_ j),\) \(n^*:=n-s\). In the formulae proved in Theorems 3.2 and 3.3 it is certain only that \(n^*\leq n-s\), but for every \(\{d_ j\}\) all the coefficients \(\beta_ j\) are calculated from a triangular system of equations (the \(\gamma_ j's\) too). Theorems 3.2 (except for the simplest case) and 2.4 have no analogues in Padé approximation.
scientific article; zbMATH DE number 4013227