Abstract
In this paper, we provide a brief overview of several refinements and applications of the Markov-type inequalities in various contexts.
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References
Baouendi, M.S., Goulaouic C.: Approximation polynomiale de fonctions C ∞ et analytiques. Ann. Inst. Fourier Grenoble 21, 149–173 (1971)
Baran, M.: Markov inequality on sets with polynomial parametrization. Ann. Polon. Math. 60(1), 69–79 (1994)
Baran, M., Pleśniak, W.: Markov’s exponent of compact sets in C n. Proc. Am. Math. Soc. 123(9), 2785–2791 (1995)
Baran, M., Pleśniak, W.: Polynomial inequalities on algebraic sets. Studia Math. 141(3), 209–219 (2000)
Baran, M., Pleśniak, W.: Characterization of compact subsets of algebraic varieties in terms of Bernstein type inequalities. Studia Math. 141(3), 221–234 (2000)
Bedford, E., Taylor, B.A.: A new capacity for plurisubharmonic functions. Acta Math. 149, 1–40 (1982)
Benko, D., Erdélyi, T.: Markov Inequality for polynomials of degree n with m distinct zeros. J. Approx. Theory 122(2), 241–248 (2003)
Bernstein, S.N.: Sur Lordre de la meilleure approximation des fonctions continues par des polynômes de degré donné. Mémoires de lAcadémie Royale de Belgique 4, 1–103 (1912)
Bernstein, S.N.: Collected Works: Vol. I, Constr. Theory of Functions (1905–1930), English Translation, Atomic Energy Commission, Springfield, VA (1958)
Białas, L., Volberg, A.: Markov’s property of the Cantor ternary set. Studia Math. 104, 259–268 (1993)
Białas-Cież, L.: Equivalence of Markov’s property and Hölder continuity of the Green function for Cantor-type sets. East J. Approx, 1(2), 249–253 (1995)
Białas-Cież,, L.: Markov sets in C are not polar, Jagiellonian University (1996)
Bierstone, E.: Extension of Whitney fields from subanalytic sets. Invent. Math. 46, 277–300 (1978)
Bierstone, E.: Differentiable functions. Bol. Soc. Bras. Mat. 12(2), 139–190 (1980)
Bierstone, E., Milman, P.D.: Semianalytic and subanalytic sets. Institut des Hautes Études Scientifiques, Publications Mathématiques 67, 5–42 (1988)
Boas, R. P.: Inequalities for the derivatives of polynomials. Math. Mag. 42, 165–174 (1969)
Borwein, P.: Markovs Inequality for Polynomials with Real Zeros. Proc. Am. Math. Soc. 93(1), 43–47 (1985)
Borwein, P., Erdélyi, T.: Markov and Bernstein type inequalities on subsets of [−1, 1] and [ −π, π]. Acta Math. Hungar. 65, 189–194 (1994)
Borwein, P., Erdélyi, T.: Markov and Bernstein type inequalities in L p for classes of polynomials with constraints. J. Lond. Math. Soc. 51(2), 573–588 (1995)
Borwein, P., Erdélyi, T.: Polynomials and Polynomial Inequalities. Springer, New York (1995)
Borwein, P.B., Erdélyi, T.: Markov- and Bernstein-type inequalities for polynomials with restricted coefficients. Ramanujan J. 1, 309–323 (1997)
Bos, L., Levenberg, N., Milman, P., Taylor, B.A.: Tangential Markov inequalities characterize algebraic submanifolds of R N. Indiana Univ. Math. J. 44(1), 115–138 (1995)
Coatmelec, C: Approximation et interpolation des fonctions différentiables des plusieurs variables. Ann. Sci. École Norm. Sup. 83(3), 271–341 (1966)
Daras, N.J.: Generalized Padé-type approximants to continuous functions. Anal. Math. 31, 251–268 (2005)
Dineen, S.: Complex Analysis in Locally Convex Spaces. North-Holland, Amsterdam (1981)
Dryanov, D., Fournier, R.: Bernstein and Markov type inequalities. Preprint CRM -2929, Centre de Recherches Mathématiques, Université de Montréal, (2003). See also, Dryanov, D., Fournier, R.: Some extensions of the Markov inequality for polynomials. Preprint CRM -3122, Centre de Recherches Mathématiques, Université de Montréal (2004)
Duffin, R.J., Schaeffer, A.C.: A refinement of an inequality of the brothers Markoff. Trans. Am. Math. Soc. 50, 517–528 (1941)
Duffin, R.J., Schaeffer, A.C.: Commentary on problems 73 and 74. In: Mauldin, R.D. (ed.) The Scottish Book, pp. 143–150. Birkhäuser, Basel (1981)
Erdélyi, T.: Markov- and Bernstein-type inequalities for Müntz polynomials and exponential sums in L p. J. Approx. Theory 104(1), 142–152 (2000)
Erdélyi, T., Kroó, A.: Markov-type inequalities on certain irrational arcs and domains. J. Approx. Theory 130(2), 113–124 (2004)
Erdélyi, T., Kroó, A., Szabados, J.: Markov-Bernstein type inequalities on compact subsets of R. Anal. Math. 26 17–34 (2000)
Erdös, P.: On extremal properties of the derivatives of polynomials. Ann. Math. 41(2), 310–313 (1940)
Eremenko, A.: A Markov-type inequality for arbitrary plane continua. Proc. Am. Math. Soc. 135, 1505–1510 (2007)
Frappier, C.: Quelques problémes extrémaux pour les polynômes at les fonctions entieées de type exponentiel. Ph.D. Dissertation Université de Montréal (1982)
Frappier, C., Rahman, Q.I., Ruscheweyh, St.: New inequalities for polynomials. Trans. Am. Math. Soc. 288, 69–99 (1985)
Goetgheluck, P.: Inégalité de Markov dans les ensembles efillés. J. Approx. Theory 30, 149–154 (1980)
Goetgheluck, P.: Polynomial inequalities on general subsets of R N. Colloq. Math. 57, 127–136 (1989)
Goetgheluck, P.: On the Markov inequality in L p-spaces. J. Approx. Theory 62, 197–205 (1990)
Goetgheluck, P., Pleśniak, W.: Counter-examples to Markov and Bernstein Inequalities. J. Approx. Theory 69, 318–325 (1992)
Harris, L.A.: Bounds on the derivatives of holomorphic functions of vectors. Proc. Colloq. Anal. Rio de Janeiro (1972), 145–163, Act. Sci. et Ind., Hermann, Paris (1975)
Harris, L.A.: Markovs inequality for polynomials on normed linear spaces. Math. Balkanica N. S. 16, 315–326 (2002)
Hille, E., Phillips, R.S.: Functional Analysis and Semi-Groups. Am. Math. Soc. Colloq. Publ. 31, AMS, Providence (1957)
Hille, E., Szegö, G., Tamarkin, J.D.: On some generalizations of a theorem of A. Markoff. Duke Math. J. 3, 729–739 (1937)
Jonsson, A.: Markov’s inequality on compact sets. In: Brezinski, C., Gori, L., Ronveaux, A. (eds.) Orthogonal Polynomials and Their Applications, pp. 309–313. Baltzer, Basel (1991)
Kellogg, O.D.: On bounded polynomials in several variables. Math. Z. 27, 55–64 (1928)
Kosek, M.: Hölder continuity property of filled-in Julia sets in C n. Proc. Am. Math. Soc. 125(7), 2029–2032 (1997)
Kroó, A., Révész, S.: On Bernstein and Markov-type inequalities for multivariate polynomials on convex bodies. J. Approx. Theory 99, 134–152 (1999)
Lorentz, G.G.: The degree of approximation by polynomials with positive coefficients. Math. Ann. 151, 239–251 (1963)
Łojasiewicz, S.: Ensembles semi-analytiques. Institut des Hautes Études Scientifiques, Bures-sur-Yvette (1964)
Markov, A.A.: On a problem of D. I. Mendeleev. Zap. Im. Akad. Nauk. 62, 1–24 (1889)
Markov, V.: Über die Funktionen, die in einem gegebenen Intervall möglichst wenig von Null abweichen. Math. Ann. 77, 213–258 (1916)
Milovanovic, G.V., Mitrinovic, D.S. Rassias, Th.M.: Topics in Polynomials, Extremal Problems, Inequalities, Zeros. World Scientific, Singapore (1994)
Milówka, B.: Markovs inequality in Banach algebras. In: 5th Summer School in Potential Theory, Kraków (2006)
Mityagin, B.S.: Approximate dimension and bases in nuclear spaces. Russ. Math. Surv. 16, 59–127 (1961)
Muñoz, G., Sarantopoulos, Y.: Bernstein and Markov-type inequalities for polynomials on real Banach spaces. Math. Proc. Camb. Phil. Soc. 133(3), 515–530 (2002)
Nadzhmiddinov, D., Subbotin, Y.N.: Markov inequalities for polynomials on triangles. Mat. Zametki 46 (1989); English translation, Math. Notes 627–631 (1990)
Newman, P.D.J.: Derivative bounds for Müntz polynomials. J. Approx. Theory 18, 360–362 (1976)
Pawlucki, W., Pleśniak, W.: Markov’s inequality and C ∞ functions on sets with polynomial cusps. Math. Ann. 275, 467–480 (1986)
Pawlucki, W., Pleśniak, W.: Extension of C ∞ functions from sets with polynomial cusps. Studia Math. 88, 279–287 (1988)
Pleśniak, W.: A Cantor regular set which does not have Markov’s property. Ann. Polon. Math. 51, 269–274 (1990)
Pleśniak, W.: Recent progress in multivariate Markov inequality. Approximation Theory, In Memory of A. K. Varma, pp. 449–464. Marcel Dekker, New York (1998)
Pleśniak, W.: Markov’s inequality and the existence of an extension operator for C ∞ functions. J. Approx. Theory 61, 106–117 (1990)
Pleśniak, W.: Extension and polynomial approximation of ultradifferentiable functions in R n. Bull. Soc. Roy. Sci. Liége 63(5), 393–402 (1994)
Pleśniak, W.: Inegalités de Markov en plusieurs variables. Int. J. Math. Math. Sci. 14, Article ID 24549, 1–12 (2006)
Pommerenke, Ch.: On the derivative of a polynomial. Michigan Math. J. 6, 373–375 (1959)
Sarantopoulos, Y.: Bounds on the derivatives of polynomials on Banach spaces. Math. Proc. Camb. Philos. Soc. 110, 307–312 (1991)
Seeley, R.T.: Extension of C ∞ functions defined on a half-space. Proc. Am. Math. Soc. 15, 625–626 (1964)
Siciak, J.: Degree of convergence of some sequences in the conformal mapping theory. Colloq. Math. 16, 49–59 (1967)
Siciak, J.: Extremal plurisubharmonic functions in C n. Ann. Pol. Math. 39, 175–211 (1981)
Siciak, J.: Highly non continuable functions on polynomially convex sets. Univ. Iagello Acta Math. 29, 95–107 (1985)
Siciak, J.: Rapid polynomial approximation on compact sets in C n. Univ. Iagello. Acta Math. 30, 145–154 (1993)
Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)
Szabados, J.: Bernstein and Markov type estimates for the derivative of a polynomial with real zeros. In: Butzer, P.L., Sz-Nagy, B., Gorlick, E. (eds.) Functional Analysis and Approximation, pp. 177-188. Birkhauser, Basel (1981)
Szabados, J., Varma, A.K.: Inequalities for derivatives of polynomials having real zeros. In: Cheney, E.W. (ed.) Approximations Theory III, pp. 881–888. Academic Press, New York (1980)
Toókos, F., Totik, V.: Markov inequality and Green functions. Rendiconti del Circolo Matematico di Palermo II 76, 91–102 (2005)
Totik, V.: Markov constants for Cantor sets. Acta Sci. Math. Szeged 60, 715–734 (1995)
Wilhelmsen, D.R.: A Markov inequality in several dimensions. J. Approx. Theory 11, 216–220 (1974)
Whitney, H.: Analytic extension of differentiable functions defined in closed sets. Trans. Am. Math. Soc. 36, 63–89 (1934)
Zeriahi, A.: Inégalités de Markov et développement en série de polynômes orthogonaux des fonctions C ∞ et A ∞. In: Fornaess, J.F. (ed.) Proceedings of the Special Year of Complex Analysis of the Mittag-Leffler Institute 1987–1988, pp. 693–701. Princeton University Press, Princeton (1993)
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Daras, N.J. (2014). Markov-Type Inequalities with Applications in Multivariate Approximation Theory. In: Rassias, T., Tóth, L. (eds) Topics in Mathematical Analysis and Applications. Springer Optimization and Its Applications, vol 94. Springer, Cham. https://doi.org/10.1007/978-3-319-06554-0_11
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