The bounding properties of the multipoint Padé approximant to a series of Stieltjes on the real line

Abstract

The multipoint Padé approximants to a series of Stieltjes and the corresponding complementary approximants are defined. Their bounding properties are established, and those of the usual [n,n−1] and [n,n] Padé approximants are shown to be special cases. It is further shown how information in the form of the first few terms in the expansion of the series of Stieltjes in inverse powers of x may be incorporated into the multipoint Padé approximant and its complement, thereby tightening the bounds supplied by the approximants. The usual ``two‐point'' Padé approximant occurs as a special case of following this procedure. Physically occurring series of Stieltjes can often be written in the form ${\mathrm{G(x)=\Sigma }}_{\text{k\hspace{0.17em}=\hspace{0.17em}1}}^K{\mathrm{\hspace{0.17em}V}}_k\mathrm{/(}{\mathcal{E}}_k\mathrm{+x)+H(x),}$ where G(x) is a series of Stieltjes with radius of convergence R > 0, H(x) is a series of Stieltjes with radius of convergence R′ > R, $\mathrm{R⩽}{\mathcal{E}}_1<{\mathcal{E}}_2\mathrm{<\dots <}{\mathcal{E}}_K\mathrm{<R\prime ,}$ and 0 < V_k < ∞ (k = 1,2,…, k). In addition to the bounds supplied by the multipoint Padé approximants for X ∈ (−R, ∞), it is shown that the approximants also exhibit interesting bounding properties for X ∈ (−R′, −R). A theorem on these bounding properties is proved. It is further shown that the multipoint Padé approximants yield best possible upper bounds on the ${\mathcal{E}}_k$ and on V₁, but, in general, do not yield straightforward bounds on V₂, V₃, …, V_K. Finally, the effect of fixing the locations of the poles of the multipoint Padé approximant and its complement at the correct values $\mathrm{x\hspace{0.17em}=\hspace{0.17em}-}{\mathcal{E}}_k\text{\hspace{0.17em}(k\hspace{0.17em}=\hspace{0.17em}1,2,…,K)}$ is considered. The resulting approximants then impose a complementary pair of bounds on G(x) for −R′ < x < ∞, which in most cases will be the best possible. In particular, one can now usually obtain best possible upper and lower bounds on V₁, V₂,…, V_K.