zbMATH — the first resource for mathematics

Examples
Geometry Search for the term Geometry in any field. Queries are case-independent.
Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact.
"Topological group" Phrases (multi-words) should be set in "straight quotation marks".
au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted.
Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff.
"Quasi* map*" py: 1989 The resulting documents have publication year 1989.
so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14.
"Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic.
dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles.
py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses).
la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

Operators
a & b logic and
a | b logic or
!ab logic not
abc* right wildcard
"ab c" phrase
(ab c) parentheses
Fields
any anywhere an internal document identifier
au author, editor ai internal author identifier
ti title la language
so source ab review, abstract
py publication year rv reviewer
cc MSC code ut uncontrolled term
dt document type (j: journal article; b: book; a: book article)
Poles and zeros of best rational approximants of |x|. (English) Zbl 0815.41015

Let r n * =p/q (where p and q are polynomials of degree at most n, with real coefficients) be a best rational approximation in the Chebyshev sense (sup-norm) of |x| on [-1,1] and E n,n =E n,n (|x|,[-1,1]) – the minimal approximation error.

In a previous paper the author has proved an important result, conjectured early by Varga et al., namely the strong asymptotic error formula: lim n e πn E n,n (|x|,[-1,1])=8. The present paper completes the above one by an exhaustive analysis of an asymptotic behaviour of poles and zeros of r n * and of the extreme points of error function |x|-r n * . More precisely, the author gives, for the large n, the formulas are individual location of the above points. Let r n * =p/q (where p and q are polynomials of degree at most n, with real coefficients) be a best rational approximation in the Chebyshev sense (sup-norm) of |x| on [-1,1] and E n,n =E n,n (|x|,[-1·1]) – the minimal approximation error. In a previous paper the author has proved an important result, conjectured early by Varga et al., namely the strong asymptotic error formula: lim n e πn E n,n (|x|,[-1,1])=8. The present paper completes the above one by an exhaustive analysis of an asymptotic behaviour of poles and zeros of r* n and of the extreme points of error function |x|-r n * . More precisely, the author gives, for the large n, the formulas for individual location of the above points.

MSC:
41A20Approximation by rational functions
41A25Rate of convergence, degree of approximation
41A44Best constants (approximations and expansions)
References:
[1][BIS]H.-P. Blatt, A. Iserles, E. B. Saff (1987):Remarks on the behavior of zeros and poles of best approximating polynomials and rational functions. In: Algorithms for Approximation (J. C. Mason, M. G. Cox, eds.). Institute of Mathematics and Its Applications Conference Series, vol. 10. Oxford: Clarendon Press, pp. 437-445.
[2][BS]H.-P. Blatt, E. B. Saff (1986):Behavior of zeros of polynomials of near best approximation. J. Approx. Theory,46:323-344. · Zbl 0605.41026 · doi:10.1016/0021-9045(86)90069-9
[3][Bu]A. P. Bulanov (1968):Asymptotics for the least deviation of |x| from rational functions. Mat. Sb.,76(118):288-303 (English translation (1968): Math USSR-SB,5:275-290).
[4][Ga]T. Ganelius (1979):Rational approximation of x ? on [0, 1]. Anal. Math.,5:19-33. · Zbl 0425.41017 · doi:10.1007/BF02079347
[5][Go]G. M. Golusin (1957): Geometrische Funktionentheorie. Berlin: Deutscher Verlag der Wissenschaften.
[6][GR]I. S. Gradshteyn, I. M. Ryshik (1980): Table of Integrals, Series, and Products. New York: Academic Press.
[7][HK]W. K. Hayman, P. B. Kennedy (1976): Subharmonic Functions. London: Academic Press.
[8][La]N. S. Landkof (1972): Foundations of Modern Potential Theory. Grundlehren der Mathematischen Wissenschaften, vol. 190. New York: Springer-Verlag.
[9][Me]G. Meinardus (1967): Approximation of Functions: Theory and Numerical Methods. New York: Springer-Verlag.
[10][Ne]D. J. Newman (1964):Rational approximation to |x|. Michigan Math. J.,11:11-14. · Zbl 0138.04402 · doi:10.1307/mmj/1028999029
[11][St1]H. Stahl (1992):Best uniform rational approximation of |x| on [?1, 1]. Mat. Sb.,183:85-118.
[12][St2]H. Stahl (1992):Uniform approximation of |x| In: Methods of Approximation Theory in Complex Analysis and Mathematical Physics, (A. A. Gonchar, E. B. Saff, eds.). Moscow: Nauka, pp. 110-130.
[13][Va]R. S. Varga (1990): Scientific Computation on Mathematical Problems and Conjectures. CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 60. Philadelphia: CBMS.
[14][Vj1]N. S. Vjacheslavov (1975):On the uniform approximation of |x| by rational functions. Dokl. Adad. Nauk SSSR,220:512-515 (English translation 1975: Soviet Math. Dokl.,16: 100-104).
[15][Vj2]N. S. Vjacheslavov (1980):On the approximation of x ? by rational functions. Izv. Akad. Nauk SSSR Ser. Mat.,44:92-109 (English translation (1981): Math. USSR. Izv.,16:83-101).
[16][VRC]R. S. Varga, A. Ruttan, A. J. Carpenter (1991):Numerical results on best uniform rational approximation of |x| on [?1,1]. Mat. Sb.,182(11):1523-1541 (English translation (1992): Math. USSR-Sb.,74(2):271-290).