Browse Prior Art Database

SQUARE BLOCKS AND EQUIOSCILLATION IN THE PADS, WALSH, AND CF TABLES

IP.com Disclosure Number: IPCOM000128193D
Original Publication Date: 1984-Dec-31
Included in the Prior Art Database: 2005-Sep-15
Document File: 11 page(s) / 34K

Publishing Venue

Software Patent Institute

Related People

Lloyd N. Trefethen: AUTHOR [+3]

Abstract

It is well known that degeneracies in the form of repeated entries always occupy square blocks in the Pade table, and likewise in the Walsh table of real rational Chebyshev approximants on an interval. The same is true in complex CF (Caratheodory-Fej6r) approximation on a circle. We show that these block structure results have a common ori- gin in the existence of equioscillation-type characterization theorems for each of these three approximation problems. Consideration of posi- tion within a block is then shown to be a fruitful guide to various questions whose answers are affected by degeneracy.

This text was extracted from a PDF file.
This is the abbreviated version, containing approximately 14% of the total text.

Page 1 of 11

THIS DOCUMENT IS AN APPROXIMATE REPRESENTATION OF THE ORIGINAL.

SQUARE BLOCKS AND EQUIOSCILLATION IN THE PADS, WALSH, AND CF TABLES

BY

Lloyd N. Trefethen~

Report ~f 1 i0

Supported by an NSF Postdoctoral Fellowship and by the U.S. Department of Energy under contract DE-AC02-76-ER03077-V. SQUARE BLOCKS AND EQUIOSCILLATION IN THE PADE, WALSH, AND CF TABLES

Lloyd N. Trefethen* Courant Institute of Mathematical Sciences New York University New York, NY 10012

Abstract.

It is well known that degeneracies in the form of repeated entries always occupy square blocks in the Pade table, and likewise in the Walsh table of real rational Chebyshev approximants on an interval. The same is true in complex CF (Caratheodory-Fej6r) approximation on a circle. We show that these block structure results have a common ori- gin in the existence of equioscillation-type characterization theorems for each of these three approximation problems. Consideration of posi- tion within a block is then shown to be a fruitful guide to various questions whose answers are affected by degeneracy.

0. Introduction

Consider the following three problems in rational approximation. In each case m and n are nonnegative integers, except that m may be negative in the CF case.

CHEBYSHEV ("T"). Let f be real and continuous on I = [-1,1), and let R~ be the set of rational functions of type (m,n) with real coefficients. Problem: find r* a Rr ,;n such that

(Equation Omitted)

where ~~*~I is the supremum norm on I .

" PADE ("P"). Let f be a complex formal power series in z , and let Rmn be the set of rational functions of type (m,n) with complex

coefficients. Problem: find rp a Rmn such that

(Equation Omitted)

New York University Page 1 Dec 31, 1984

Page 2 of 11

SQUARE BLOCKS AND EQUIOSCILLATION IN THE PADS, WALSH, AND CF TABLES

*Supported by an NSF Postdoctoral Fellowship and by the U.S. Dept. of Energy under contract DE-AC02-76-ER03077-V.

(Equation Omitted)

Let f be a continuous function on

(Equation Omitted)

whose Fourier series converges absolutely. Let Ian be the set of "extended rational" functions representable in the form

(Equation Omitted)

where q has all of its zeros in JzJ > 1 , and the series for p converges there and is bounded except possibly near z = ~ . Problem: find r* a Amn such that

(Equation Omitted)

where

(Equation Omitted)

norm on S .

See [12] for information on Chebvshev approximation, [1,6] for Pads, and [7,14,15] for CF. (The CF approximant defined above is act-ually the "extended CF approximant"; in practice it would be projected onto a function rcf a Rmn to yield a near-best Chebyshev approximant on a disk.) All three problems have unique solutions, and these can be constructed numerically: rp by solving a finite Hankel system of linear equations, r* by a procedure such as the Remes algorithm, and r* via a singular value decomposition of an infinite Hankel matrix of Laurent series coefficients. We will not go into this. The Pads table is the array obtained by...