Square blocks and equioscillation in the Padé, walsh, and cf tables
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It is well known that degeneracies in the form of repeated entries always occupy square blocks in the Padé table, and likewise in the Walsh table of real rational Chebyshev approximants on an interval. The same is true in complex CF (Carathéodory-Fejér) approximation on a circle. We show that these block structure results have a common origin in the existence of equioscillation-type characterization theorems for each of these three approximation problems. Consideration of position within a block is then shown to be a fruitful guide to various questions whose answers are affected by degeneracy.
KeywordsBlock Structure Error Curve Chebyshev Approximation Block Pattern Pade Approximation
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- G. Baker and P. Graves-Morris, Padé Approximants (2 vols.), Encyc. of Math. v. 13 and 14, Addison-Wesley, 1981.Google Scholar
- A. Bultheel, paper in this volume.Google Scholar
- M. H. Gutknecht, On complex rational approximation II, in Computational Aspects of Complex Analysis, H. Werner et al. (eds.), D. Reidel, Dordrecht/Boston/Lancaster, 1983.Google Scholar
- M. H. Gutknecht, E. Hayashi, and L. N. Trefethen, The CF table, in preparation.Google Scholar
- A. Magnus, private communication, 1983.Google Scholar
- G. Meinardus, Approximation of Functions: Theory and Numerical Methods, Springer, 1967.Google Scholar
- L. N. Trefethen, Chebyshev approximation on the unit disk, in Computational Aspects of Complex Analysis, H. Werner et al. (eds.), D. Reidel, Dordrecht/Boston/Lancaster, 1983.Google Scholar
- L. N. Trefethen and M. H. Gutknecht, On convergence and degeneracy in rational Padé and Chebyshev approximation, SIAM J. Math. Anal., to appear.Google Scholar