Convergence Properties Related to p-Point Padé Approximants of Stieltjes Transforms
✍ Scribed by O. Njastad
- Publisher
- Elsevier Science
- Year
- 1993
- Tongue
- English
- Weight
- 377 KB
- Volume
- 73
- Category
- Article
- ISSN
- 0021-9045
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✦ Synopsis
Let (\psi) be a finite positive measure on (\mathbf{R}), and let (F_{\psi}(z)=\int_{-x}^{x}(d \psi(t) /(z-t))) be its Stieltjes transform. A special multipoint Padé approximation problem for (F_{\psi}(z)) is studied, where the interpolation points are a finite number of points (a_{1}, \ldots, a_{p}) in (\mathbf{R}) repeated cyclically and the support of (\psi) is contained in an interval bounded by adjacent interpolation points. For the case (p=3) monotone convergence of each of the subsequences (\left{P_{3 q+m}(z) / Q_{3 q+m}(z)\right}, m=0,1,2), of the multipoint Padé approximants (\left{P_{n}(z) / Q_{n}(z)\right}) is established, and sufficient conditions (involving general moments (\left.c_{f^{(i)}}=\int_{-x}^{x}\left(d \psi(t) /\left(1-a_{i}\right)^{j}\right)\right)) for divergence of the series (\sum_{q=1}^{x}\left|Q_{3 q+m}(z)\right|^{2}) are given. 1993 Academic Press, Inc.