Geodesic
On the Behavior of Zeros of Polynomials of Best and Near-Best Approximation
Canadian journal of mathematics, Tome 43 (1991) no. 5, pp. 1010-1021

Voir la notice de l'article provenant de la source Cambridge University Press

Assume ƒ is continuous on the closed disk D 1 : |z| ≤ 1, analytic in |z| ≤ 1, but not analytic on D1 . Our concern is with the behavior of the zeros of the polynomials of best uniform approximation to ƒ on D1. It is known that, for such ƒ, every point of the circle |z| = 1 is a cluster point of the set of all zeros of Here we show that this property need not hold for every subsequence of the Specifically, there exists such an f for which the zeros of a suitable subsequence all tend to infinity. Further, for near-best polynomial approximants, we show that this behavior can occur for the whole sequence. Our examples can be modified to apply to approximation in the Lq -norm on |z|= 1 and to uniform approximation on general planar sets (including real intervals).
DOI : 10.4153/CJM-1991-057-7
Mots-clés : 41A20 
Ivanov, K. G.; Saff, E. B.; Totik, V. On the Behavior of Zeros of Polynomials of Best and Near-Best Approximation. Canadian journal of mathematics, Tome 43 (1991) no. 5, pp. 1010-1021. doi: 10.4153/CJM-1991-057-7
@article{10_4153_CJM_1991_057_7,
     author = {Ivanov, K. G. and Saff, E. B. and Totik, V.},
     title = {On the {Behavior} of {Zeros} of {Polynomials} of {Best} and {Near-Best} {Approximation}},
     journal = {Canadian journal of mathematics},
     pages = {1010--1021},
     year = {1991},
     volume = {43},
     number = {5},
     doi = {10.4153/CJM-1991-057-7},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CJM-1991-057-7/}
}
TY  - JOUR
AU  - Ivanov, K. G.
AU  - Saff, E. B.
AU  - Totik, V.
TI  - On the Behavior of Zeros of Polynomials of Best and Near-Best Approximation
JO  - Canadian journal of mathematics
PY  - 1991
SP  - 1010
EP  - 1021
VL  - 43
IS  - 5
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CJM-1991-057-7/
DO  - 10.4153/CJM-1991-057-7
ID  - 10_4153_CJM_1991_057_7
ER  - 
%0 Journal Article
%A Ivanov, K. G.
%A Saff, E. B.
%A Totik, V.
%T On the Behavior of Zeros of Polynomials of Best and Near-Best Approximation
%J Canadian journal of mathematics
%D 1991
%P 1010-1021
%V 43
%N 5
%U http://geodesic.mathdoc.fr/articles/10.4153/CJM-1991-057-7/
%R 10.4153/CJM-1991-057-7
%F 10_4153_CJM_1991_057_7

[1] 1. Blatt, H.-P. and Saff, E.B., Behavior of zeros of polynomials of near best approximation, J. Approx. Theory 46 (1986),323–344. Google Scholar

[2] 2. Blatt, H.-P., Saff, E.B. and Simkani, M., Jentzsch-Szeg ő type theorems for the zeros of best approximants, J. London Math. Soc. 38 (1988), 307–316. Google Scholar

[3] 3. Feinerman, R.P. and Newman, D.J., Polynomial approximation. The Williams & Wilkins Company, Baltimore, 1974. Google Scholar

[4] 4. Grothmann, R. and Saff, E.B., On the behavior of zeros and poles of best uniform polynomial and rational approximation. In: Nonlinear Numerical Methods and Rational Approximations, (A. Cuyt, éd.), Reidel Publishing Company, 1988, 1988–57. Google Scholar

[5] 5. Hille, E., Analytic function theory, vol. II. Ginn and Company, Boston, 1962. Google Scholar

[6] 6. Jentzsch, R., Untersuchungen zur Théorie Analytischer Funktionen. Inaugural-dissertation, Berlin, 1914. Google Scholar

[7] 7. Jentzsch, R., Fortgesetzte Untersuchungen iiber die Abschnitte von Potenzreihen, Acta Math. 41 (1918), 253- 270. Google Scholar

[8] 8. Saff, E.B., A principle of contamination in best polynomial approximation. In: Approximation and Optimization (Gomez et al., eds.), Lecture Notes in Math., 1354, Springer-Verlag, Heidelberg, 1988,79–97. Google Scholar

[9] 9. Timan, A.F., Theory of approximation of functions of a real variable. Pergamon Press, New York, 1963. Google Scholar

[10] 10. Walsh, J.L., Interpolation and approximation by rational functions in the complex domain. AMS Colloquium Publications, XX, third edition, Amer. Math. Soc, Providence, 1960. Google Scholar

Cité par Sources :