The Density of Alternation Points in Rational Approximation Author(s): P. B. Borwein, A. Kroó, R. Grothmann, E. B. Saff Source: Proceedings of the American Mathematical Society, Vol. 105, No. 4, (Apr., 1989), pp. 881-888 Published by: American Mathematical Society Stable URL: http://www.jstor.org/stable/2047047 Accessed: 23/05/2008 18:10 Your use of the JSTOR archive indicates your acceptance of JSTOR's Terms and Conditions of Use, available at http://www.jstor.org/page/info/about/policies/terms.jsp. JSTOR's Terms and Conditions of Use provides, in part, that unless you have obtained prior permission, you may not download an entire issue of a journal or multiple copies of articles, and you may use content in the JSTOR archive only for your personal, non-commercial use. Please contact the publisher regarding any further use of this work. Publisher contact information may be obtained at http://www.jstor.org/action/showPublisher?publisherCode=ams. Each copy of any part of a JSTOR transmission must contain the same copyright notice that appears on the screen or printed page of such transmission.
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PROCEEDINGSOF THE AMERICANMATHEMATICAL SOCIETY Volume 105, Number4, April 1989
THE DENSITY OF ALTERNATION POINTS IN RATIONAL APPROXIMATION P. B. BORWEIN,
*
A. KROO,
*
R. GROTHMANN * AND E. B. SAFF
**
(Communicated by Irwin Kra)
We investigate the behavior of the equioscillation (alternation) points for the error in best uniform rational approximation on [-1, 1]. In the context of the Walsh table (in which the best rational approximant with numerator degree < m , denominator degree < n, is displayed in the nth row and the mth column), we show that these points are dense in [-1, 1], if one goes down the table along a ray above the main diagonal (n = [cm], c < 1). A counterexample is provided showing that this may not be true for a subdiagonal of the table. In addition, a Kadec-type result on the distribution of the equioscillation points is obtained for asymptotically horizontal paths in the Walsh table. ABSTRACT.
1. STATEMENT OF RESULTS
Denote by 3fm n the rational functions with numeratorin rlm' the set of algebraicreal polynomials of degreeat most m, and denominatorin rln. Then the best approximation rm ,n = Pm nqmn in SSmn to fE C[-1,1] with respect to the uniform norm (1.1)
ulghl[-l ,1]_:=
sup{lg(x)I:x
E [-1
, 1]}
is unique and is characterizedby an equioscillation property [M], i.e., there are m + n +2- d points (1.2)
-1
