Approximation of a function given by its Laurent series
Let \(f(z)\) be a function holomorphic in an annulus \(\{z: r<|z|< \mathbb{R}\}\), given by its Laurent series, and \(p\), \(p'\), \(q\) nonnegative integers. Following the spirit of the weak Padé approximation, for a fixed integer \(n\) a rational approximating function \[ [ n,p', p,q ](x)= {{N(z)} \over {Q(z)}}, \qquad N(z)= \sum^p_{k= -p'} a_k z^k, \qquad Q(z)= \sum^q_{k=0} b_k z^k, \] is constructed in such a way that the Laurent expansion of \((Qf- N) (z)\) has no terms from power \(-n- p'\) to power \(p+ q-n\). Several good qualitative properties of these approximants are proved. Thus, in the case when \(f\) is meromorphic in a larger annulus, an analogue of the Montessus de Mallore theorem is established (for \(p, p'\to \infty\)), along with relations similar to the relations of the QD algorithm. Moreover, the denominators \(Q(z)\) are linked to reverse vector orthogonal polynomials, and recurrence relations for both \(N\) and \(Q\) are presented.
scientific article; zbMATH DE number 871292