Block Structure

Rational Approximation and Interpolation

Volume 1105 of the series Lecture Notes in Mathematics pp 170-181


Square blocks and equioscillation in the Padé, walsh, and cf tables

  • Lloyd N. TrefethenAffiliated withCourant Institute of Mathematical Sciences, New York University

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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.