Multidimensional analogues of the Markov and Bernstein inequalities
Izvestiya. Mathematics , Tome 65 (2001) no. 6, pp. 1197-1241.

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We obtain estimates for the derivatives of algebraic polynomials on convex bodies and centrally symmetric convex bodies.
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V. I. Skalyga. Multidimensional analogues of the Markov and Bernstein inequalities. Izvestiya. Mathematics , Tome 65 (2001) no. 6, pp. 1197-1241. http://geodesic.mathdoc.fr/item/IM2_2001_65_6_a4/

[1] Markov A. A., “Ob odnom voprose D. I. Mendeleeva”, Izbrannye trudy, GITTL, M.–L., 1948

[2] Markov V. A., O funktsiyakh, naimenee uklonyayuschikhsya ot nulya v dannom promezhutke, SPb., 1892 | Zbl

[3] Bernshtein S. N., Sobranie sochinenii, T. 1, Izd-vo AN SSSR, M., 1952

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

[5] Voronovskaya E. V., “Funktsional pervoi proizvodnoi i utochnenie teoremy A. A. Markova”, Izv. AN SSSR. Ser. matem., 23:6 (1959), 951–962 | MR | Zbl

[6] Gusev V. A., “Funktsionaly proizvodnykh ot algebraicheskogo polinoma i teorema V. A. Markova”, Izv. AN SSSR. Ser. matem., 25:3 (1961), 367–384 | MR | Zbl

[7] Videnskii V. S., “Ob otsenkakh proizvodnykh mnogochlena”, Izv. AN SSSR. Ser. matem., 15:5 (1951), 401–420 | MR | Zbl

[8] Riesz M., Deutsche Math.-Ver., 23 (1914), 354 | Zbl

[9] Szego G., “Über einen Satz des Herrn Serge Bernstein”, Schriften der Konigsberger Gelehrten Gesellschaft, 5 (1928), 59–70 | Zbl

[10] Nikolskii S. M., “Obobschenie neravenstva S. N. Bernshteina”, DAN SSSR, 60 (1948), 1507–1510 | MR | Zbl

[11] Stechkin S. B., “Obobschenie nekotorykh neravenstv S. N. Bernshteina”, DAN SSSR, 60 (1948), 1511–1514 | MR | Zbl

[12] Privalov I. I., “Integral Koshi”, Izv. Saratov. univer. Fiz.-mat. fak., 1918, no. 1

[13] Jackson D., “On application of Markoff's theorem to problems of approximation in the complex domain”, Bull. Amer. Math. Soc., 37 (1931), 883–890 | DOI | Zbl

[14] Videnskii V. S., “Ekstremalnye otsenki proizvodnoi trigonometricheskogo polinoma na otrezke, menshem, chem period”, DAN SSSR, 130:1 (1960), 13–16 | MR | Zbl

[15] Zinger M. Ya., Elementy differentsialnoi teorii chebyshevskikh priblizhenii, Nauka, M., 1975 | MR | Zbl

[16] Telyakovskii S. A., “O rabotakh po teorii priblizheniya funktsii, vypolnennykh v MIAN”, Tr. MIAN SSSR, 182, Nauka, M., 1988, 128–179 | MR

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

[18] Ganzburg M. I., “Teoremy Dzheksona i Bernshteina v $\mathbb R^m$”, UMN, 34:1 (1979), 225–226 | MR | Zbl

[19] Babenko K. I., “Ob approksimatsii periodicheskikh funktsii neskolkikh peremennykh trigonometricheskimi polinomami”, DAN SSSR, 132 (1960), 982–985 | MR | Zbl

[20] Telyakovskii S. A., “O neravenstvakh dlya proizvodnykh trigonometricheskikh polinomov neskolkikh peremennykh”, Sib. matem. zhurn., 4 (1963), 1404–1411

[21] Temlyakov V. N., “Approksimatsiya periodicheskikh funktsii neskolkikh peremennykh s ogranichennoi smeshannoi proizvodnoi”, Tr. MIAN SSSR, 156, Nauka, M., 1980, 233–260 | MR | Zbl

[22] Temlyakov V. N., Approksimatsiya funktsii s ogranichennoi smeshannoi proizvodnoi, Tr. MIAN SSSR, 178, Nauka, M., 1986 | MR | Zbl

[23] Coatmelic C., “Approximation et interpolation des fonctions differentiables de plusieurs variables”, Ann. Sci. Ecole Norm. Sup. Ser., 83:3 (1966), 271–341 | MR

[24] Don R., “Wilhelmsen A. Markov Inequality in Several Dimensions”, J. of Approx. Theory, 11:3 (1974), 216–220 | DOI | MR | Zbl

[25] Nadzhmiddinov D., Subbotin Yu. N., “Neravenstva Markova na treugolnikakh”, Matem. zametki, 46:2 (1989), 76–82 | MR | Zbl

[26] Bialas-Ciez L., Goetgheluck P., “Constants in Markov's inequality on convex sets”, EAST J. Approx., 1:3 (1995), 379–389 | MR | Zbl

[27] Andrianov A. V., “Analogi neravenstv A. Markova i S. Bernshteina dlya mnogochlenov v banakhovykh prostranstvakh”, Matem. zametki, 52:5 (1992), 13–21 | MR | Zbl

[28] Skalyga V. I., “Analogi neravenstv Markovykh i Bernshteina dlya polinomov v banakhovykh prostranstvakh”, Izv. RAN. Ser. matem., 61:1 (1997), 141–156 | MR | Zbl

[29] Skalyga V. I., “Analogi neravenstv Markovykh i Bernshteina na vypuklykh telakh v banakhovykh prostranstvakh”, Izv. RAN. Ser. matem., 62:2 (1998), 169–192 | MR | Zbl

[30] Skalyga V. I., “Otsenki proizvodnykh polinomov na vypuklykh telakh”, Tr. MIAN, 218, Nauka, M., 1997, 374–384 | MR | Zbl

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

[32] Baran M., “Markov inequality on sets with polynomial parametrization”, Ann. Polon. Math., LX:1 (1994), 69–79 | MR

[33] Sarantopoulos Y., “Bounds on the derivatives of polynomials on Banach spaces”, Math. Proc. Camb. Phil. Soc., 110 (1991), 307–312 | DOI | MR | Zbl

[34] Kellogg O. D., “On bounded polynomials on several variables”, Math. Zeit., 27:1 (1927), 55–66 | DOI | MR

[35] The Scottish Book “Mathematics from the Scottish Cafe”, ed. R. D. Mauldin, Birkhauser, 1981 | MR

[36] Harris L. A., “Bernstein's polynomial inequalities and functional analysis”, Irish Math. Soc. Bulletin, 36 (1996), 19–33 | MR | Zbl

[37] Harris L. A., “A Bernstein–Markov theorem for normed spaces”, J. Math. Analysis, 208 (1997), 476–486 | DOI | MR | Zbl

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

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

[40] Harris L. A., “Bounds on the derivatives of holomorphic functions of vectors”, Analyse Fonctionnele et Applications, ed. Leopoldo Nachbin, Hermann, Paris, 1975, 145–163 | MR

[41] Kopec J., Musielak J., “On the estimation of the norm of the $n$-linear symmetric operation”, Studia Math., 15 (1955), 29–30 | MR | Zbl

[42] Klark F., Optimizatsiya i negladkii analiz, Nauka, M., 1988 | MR | Zbl

[43] Akhiezer N. I., Lektsii po teorii approksimatsii, OGIZ, M.–L., 1947

[44] Schaeffer A. C., Duffin R. I., “A refinement of an inequality of the brothers Markoff”, Trans. Amer. Math. Soc., 50:3 (1941), 517–528 | DOI | MR | Zbl