Geodesic
V.~A.~Markov's theorems in normed spaces
Izvestiya. Mathematics , Tome 72 (2008) no. 2, pp. 383-412.

Voir la notice de l'article provenant de la source Math-Net.Ru

We obtain multidimensional analogues of V. A. Markov's inequalities for algebraic polynomials on centrally symmetric convex bodies and an analogue of the Schaeffer–Duffin inequality for polynomials on multidimensional cubes. We establish necessary and sufficient conditions for these inequalities to become equalities, and describe the sets of extremals. 
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V. I. Skalyga. V.~A.~Markov's theorems in normed spaces. Izvestiya. Mathematics , Tome 72 (2008) no. 2, pp. 383-412. http://geodesic.mathdoc.fr/item/IM2_2008_72_2_a7/

[1] V. A. Markov, O funktsiyakh, naimenee uklonyayuschikhsya ot nulya v dannom promezhutke, Izd-vo Imp. AN, SPb., 1892

[2] A. A. Markov, Ob odnom voprose D. I. Mendeleeva. Izbrannye trudy, GITL, M.–L., 1948

[3] A. C. Schaeffer, R. J. Duffin, “On some inequalities of S. Bernstein and W. Markoff for derivatives of polynomials”, Bull. Amer. Math. Soc., 44:4 (1938), 289–297 | DOI | Zbl

[4] S. A. Telyakovskiĭ, “Work on the theory of approximation of functions carried out at the V. A. Steklov Institute of Mathematics”, Proc. Steklov Inst. Math., 182:1 (1990), 141–197 | MR | Zbl

[5] A. V. Andrianov, “On some open problems for algebraic polynomials on bounded convex bodies”, East J. Approx., 5:1 (1999), 117–123 | MR | Zbl

[6] G. G. Magaril-Ilyaev, V. M. Tikhomirov, Vypuklyi analiz i ego prilozheniya, Editorial URSS, M., 2000

[7] V. I. Skalyga, “Bounds for the derivatives of polynomials on centrally symmetric convex bodies”, Izv. Math., 69:3 (2005), 607–621 | DOI | MR | Zbl

[8] V. I. Skalyga, “Analogs of the Markov and Schaeffer–Duffin inequalities for convex bodies”, Math. Notes, 68:1 (2000), 130–134 | MR | Zbl

[9] V. I. Skalyga, “Multidimensional analogues of the Markov and Bernstein inequalities”, Izv. Math., 65:6 (2001), 1197–1241 | DOI | MR | Zbl

[10] Y. Sarantopoulos, “Bounds on derivatives of polynomials on Banach spaces”, Math. Proc. Cambridge Philos. Soc., 110:2 (1991), 307–312 | DOI | MR | Zbl

[11] O. D. Kellogg, “On bounded polynomials in several variables”, Math. Z., 27:1 (1928), 55–64 | DOI | MR

[12] G. A. Muñoz, Y. Sarantopoulos, “Bernstein and Markov-type inequalities for polynomials on a real Banach spaces”, Math. Proc. Cambridge Philos. Soc., 133:3 (2002), 515–530 | DOI | MR | Zbl

[13] Yu. N. Subbotin, Yu. S. Vasilev, “Neravenstva Markova v $\mathbb R^m$, neuluchshaemye na klasse vsekh vypuklykh kompaktnykh tel”, Dokl. RAN, 360:6 (1998), 734–735 | MR | Zbl

[14] V. I. Skalyga, “Estimates for the derivatives of polynomials on convex bodies”, Proc. Steklov Inst. Math., 3:218 (1997), 372–383 | MR | Zbl

[15] L. A. Harris, “Bounds on the derivatives of holomorphic functions of vectors”, Analyse Fonctionnele et Applications, Comptes Rendus Colloq. Analyse (Rio de Janeiro, 1972), Hermann, Paris, 1975, 145–163 | MR | Zbl

[16] A. E. Taylor, “Additions to the theory of polynomials in normed linear spaces”, Tohoku Math. J., 44 (1938), 302–318 | Zbl

[17] S. Banach, “Über homogene Polynome in ($L^2$)”, Studia Math., 7 (1938), 36–44 | Zbl

[18] V. K. Dzyadyk, Vvedenie v teoriyu ravnomernogo priblizheniya funktsii polinomami, Nauka, M., 1977 | MR | Zbl

[19] M. V. Menshikov, A. M. Revyakin, A. M. Kopylova, Yu. N. Makarov, B. S. Stechkin, Kombinatornyi analiz. Zadachi i uprazhneniya, Nauka, M., 1982 | MR