Bivariate Polynomials of Least Deviation from Zero

Bojanov, Borislav D. ; Haußmann, Werner ; Nikolov, Geno P.

Canadian journal of mathematics, Tome 53 (2001) no. 3, pp. 489-505

Résumé

Bivariate polynomials with a fixed leading term ${{x}^{m}}{{y}^{n}}$ , which deviate least from zero in the uniform or ${{L}^{2}}$ -norm on the unit disk $D$ (resp. a triangle) are given explicitly. A similar problem in ${{L}^{p}}$ , $1\,\le \,p\,\le \,\infty $ is studied on $D$ in the set of products of linear polynomials.

Détail

DOI : 10.4153/CJM-2001-021-3

Mots-clés : 41A10, 41A50, 41A63

Bibliographie

[1] [1] Achieser, N. I., Vorlesungen über Approximationstheorie. Akademie-Verlag, Berlin, 1967. Google Scholar

[2] [2] Bojanov, B., An extension of the Pizzetti formula for polyharmonic functions. Acta Math. Hungar., to appear. Google Scholar

[3] [3] Bojanov, B. and Petrova, G., Numerical integration over a disc. A new Gaussian quadrature formula. Numer. Math. 80 (1998), 39–59. Google Scholar

[4] [4] Borwein, P. and Erdélyi, T., Polynomials and Polynomial Inequalities. Springer, Berlin-Heidelberg-New York, 1995. Google Scholar

[5] [5] Braß, H., Quadraturverfahren. Vandenhoeck und Ruprecht, Göttingen, 1977. Google Scholar

[6] [6] Ehlich, H. and Zeller, K., Čebyšev–Polynome in mehreren Veränderlichen. Math. Z. 93 (1966), 142–143. Google Scholar

[7] [7] Fromm, J., L-approximation to zero. Math. Z. 151 (1976), 31–33. Google Scholar

[8] [8] Gearhart, W. B., Some Chebyshev approximations by polynomials in two variables. J. Approx. Theory 8 (1973), 195–209. Google Scholar

[9] [9] Haußmann, W. and Zeller, K., Blending interpolation and best L-approximation. Arch. Math. (Basel) 40 (1983), 545–552. Google Scholar

[10] [10] Haußmann, W. and Zeller, K., Mixed norm multivariate approximation with blending functions. In: Constructive Theory of Functions (eds. Bl. Sendov et al.), Publ. House Bulg. Acad. Sci., Sofia, 1984, 403–408. Google Scholar

[11] [11] Kroó, A., Chebyshev-type extremal problems for multivariate polynomials. East J. Approx. 5 (1999), 211–221. Google Scholar

[12] [12] Milovanović, G. V., Mitinović, D. S. and Rassias, T. M., Topics in Polynomials: Extremal Problems, Inequalities, Zeros. World Scientific Publ., River Edge, NJ, 1994. Google Scholar

[13] [13] Newman, D. J. and Xu, Y., Tchebycheff polynomials on a triangular region. Constr. Approx. 9 (1993), 543–546. Google Scholar

[14] [14] Oskolkov, K., Orthonormal systems of ridge- and radial polynomials. A construction based on symmetrization and Chebyshev type ideas. Lecture Notes, Univ. South Carolina, Columbia, 1996. Google Scholar

[15] [15] Rack, H.-J., On multivariate polynomial L-approximation to zero. Anal. Math. (Budapest) 10 (1984), 241–247. Google Scholar

[16] [16] Rack, H.-J., On multivariate polynomial L-approximation to zero and related coefficient inequalities. In: Multivariate Approximation Theory III (eds.W. Schempp et al.), Internat. Ser. Numer. Math. 75, Birkhäuser, Basel, 1985, 332–342. Google Scholar

[17] [17] Reimer, M., On multivariate polynomials of least deviation from zero on the unit ball. Math. Z. 153 (1977), 51–58. Google Scholar

[18] [18] Reimer, M., On multivariate polynomials of least deviation from zero on the unit cube. J. Approx. Theory 23 (1978), 65–69. Google Scholar

[19] [19] Reimer, M., Constructive Theory of Functions. B. I. Wissenschaftsverlag, Mannheim-Wien-Zürich, 1990. Google Scholar

[20] [20] Rivlin, T. J., The Chebyshev Polynomials. Wiley, New York-London-Sydney-Toronto, 1974. Google Scholar

[21] [21] Šac, E., Certain polynomials of least deviation in the metric L . (Russian) Tallin. Polytehn. Inst. Toimetised Seer. A No. 293 (1970), 23–26. Google Scholar

[22] [22] Sloss, J. M., Chebyshev approximation to zero. Pacific J. Math. 15 (1965), 305–313. Google Scholar