Skip to main content

Markov-Type Inequalities with Applications in Multivariate Approximation Theory

  • Chapter
  • First Online:
Topics in Mathematical Analysis and Applications

Part of the book series: Springer Optimization and Its Applications ((SOIA,volume 94))

  • 1489 Accesses

Abstract

In this paper, we provide a brief overview of several refinements and applications of the Markov-type inequalities in various contexts.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Baouendi, M.S., Goulaouic C.: Approximation polynomiale de fonctions C et analytiques. Ann. Inst. Fourier Grenoble 21, 149–173 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  2. Baran, M.: Markov inequality on sets with polynomial parametrization. Ann. Polon. Math. 60(1), 69–79 (1994)

    MathSciNet  MATH  Google Scholar 

  3. Baran, M., Pleśniak, W.: Markov’s exponent of compact sets in C n. Proc. Am. Math. Soc. 123(9), 2785–2791 (1995)

    MATH  Google Scholar 

  4. Baran, M., Pleśniak, W.: Polynomial inequalities on algebraic sets. Studia Math. 141(3), 209–219 (2000)

    MathSciNet  MATH  Google Scholar 

  5. Baran, M., Pleśniak, W.: Characterization of compact subsets of algebraic varieties in terms of Bernstein type inequalities. Studia Math. 141(3), 221–234 (2000)

    MathSciNet  MATH  Google Scholar 

  6. Bedford, E., Taylor, B.A.: A new capacity for plurisubharmonic functions. Acta Math. 149, 1–40 (1982)

    Article  MathSciNet  MATH  Google Scholar 

  7. Benko, D., Erdélyi, T.: Markov Inequality for polynomials of degree n with m distinct zeros. J. Approx. Theory 122(2), 241–248 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  8. 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)

    Google Scholar 

  9. Bernstein, S.N.: Collected Works: Vol. I, Constr. Theory of Functions (1905–1930), English Translation, Atomic Energy Commission, Springfield, VA (1958)

    Google Scholar 

  10. Białas, L., Volberg, A.: Markov’s property of the Cantor ternary set. Studia Math. 104, 259–268 (1993)

    MathSciNet  MATH  Google Scholar 

  11. 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)

    MathSciNet  MATH  Google Scholar 

  12. Białas-Cież,, L.: Markov sets in C are not polar, Jagiellonian University (1996)

    Google Scholar 

  13. Bierstone, E.: Extension of Whitney fields from subanalytic sets. Invent. Math. 46, 277–300 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  14. Bierstone, E.: Differentiable functions. Bol. Soc. Bras. Mat. 12(2), 139–190 (1980)

    Article  MathSciNet  Google Scholar 

  15. Bierstone, E., Milman, P.D.: Semianalytic and subanalytic sets. Institut des Hautes Études Scientifiques, Publications Mathématiques 67, 5–42 (1988)

    Article  MathSciNet  MATH  Google Scholar 

  16. Boas, R. P.: Inequalities for the derivatives of polynomials. Math. Mag. 42, 165–174 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  17. Borwein, P.: Markovs Inequality for Polynomials with Real Zeros. Proc. Am. Math. Soc. 93(1), 43–47 (1985)

    MATH  Google Scholar 

  18. Borwein, P., Erdélyi, T.: Markov and Bernstein type inequalities on subsets of [−1, 1] and [ −π, π]. Acta Math. Hungar. 65, 189–194 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  19. 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)

    Article  MATH  Google Scholar 

  20. Borwein, P., Erdélyi, T.: Polynomials and Polynomial Inequalities. Springer, New York (1995)

    Book  MATH  Google Scholar 

  21. Borwein, P.B., Erdélyi, T.: Markov- and Bernstein-type inequalities for polynomials with restricted coefficients. Ramanujan J. 1, 309–323 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  22. 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)

    Article  MathSciNet  MATH  Google Scholar 

  23. Coatmelec, C: Approximation et interpolation des fonctions différentiables des plusieurs variables. Ann. Sci. École Norm. Sup. 83(3), 271–341 (1966)

    MathSciNet  MATH  Google Scholar 

  24. Daras, N.J.: Generalized Padé-type approximants to continuous functions. Anal. Math. 31, 251–268 (2005)

    Article  MathSciNet  Google Scholar 

  25. Dineen, S.: Complex Analysis in Locally Convex Spaces. North-Holland, Amsterdam (1981)

    MATH  Google Scholar 

  26. 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)

    Google Scholar 

  27. Duffin, R.J., Schaeffer, A.C.: A refinement of an inequality of the brothers Markoff. Trans. Am. Math. Soc. 50, 517–528 (1941)

    Article  MathSciNet  Google Scholar 

  28. 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)

    Google Scholar 

  29. Erdélyi, T.: Markov- and Bernstein-type inequalities for Müntz polynomials and exponential sums in L p. J. Approx. Theory 104(1), 142–152 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  30. Erdélyi, T., Kroó, A.: Markov-type inequalities on certain irrational arcs and domains. J. Approx. Theory 130(2), 113–124 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  31. Erdélyi, T., Kroó, A., Szabados, J.: Markov-Bernstein type inequalities on compact subsets of R. Anal. Math. 26 17–34 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  32. Erdös, P.: On extremal properties of the derivatives of polynomials. Ann. Math. 41(2), 310–313 (1940)

    Article  Google Scholar 

  33. Eremenko, A.: A Markov-type inequality for arbitrary plane continua. Proc. Am. Math. Soc. 135, 1505–1510 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  34. Frappier, C.: Quelques problémes extrémaux pour les polynômes at les fonctions entieées de type exponentiel. Ph.D. Dissertation Université de Montréal (1982)

    Google Scholar 

  35. Frappier, C., Rahman, Q.I., Ruscheweyh, St.: New inequalities for polynomials. Trans. Am. Math. Soc. 288, 69–99 (1985)

    Google Scholar 

  36. Goetgheluck, P.: Inégalité de Markov dans les ensembles efillés. J. Approx. Theory 30, 149–154 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  37. Goetgheluck, P.: Polynomial inequalities on general subsets of R N. Colloq. Math. 57, 127–136 (1989)

    MathSciNet  MATH  Google Scholar 

  38. Goetgheluck, P.: On the Markov inequality in L p-spaces. J. Approx. Theory 62, 197–205 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  39. Goetgheluck, P., Pleśniak, W.: Counter-examples to Markov and Bernstein Inequalities. J. Approx. Theory 69, 318–325 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  40. 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)

    Google Scholar 

  41. Harris, L.A.: Markovs inequality for polynomials on normed linear spaces. Math. Balkanica N. S. 16, 315–326 (2002)

    MathSciNet  MATH  Google Scholar 

  42. Hille, E., Phillips, R.S.: Functional Analysis and Semi-Groups. Am. Math. Soc. Colloq. Publ. 31, AMS, Providence (1957)

    Google Scholar 

  43. Hille, E., Szegö, G., Tamarkin, J.D.: On some generalizations of a theorem of A. Markoff. Duke Math. J. 3, 729–739 (1937)

    Article  Google Scholar 

  44. 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)

    Google Scholar 

  45. Kellogg, O.D.: On bounded polynomials in several variables. Math. Z. 27, 55–64 (1928)

    Article  MathSciNet  Google Scholar 

  46. Kosek, M.: Hölder continuity property of filled-in Julia sets in C n. Proc. Am. Math. Soc. 125(7), 2029–2032 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  47. Kroó, A., Révész, S.: On Bernstein and Markov-type inequalities for multivariate polynomials on convex bodies. J. Approx. Theory 99, 134–152 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  48. Lorentz, G.G.: The degree of approximation by polynomials with positive coefficients. Math. Ann. 151, 239–251 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  49. Łojasiewicz, S.: Ensembles semi-analytiques. Institut des Hautes Études Scientifiques, Bures-sur-Yvette (1964)

    Google Scholar 

  50. Markov, A.A.: On a problem of D. I. Mendeleev. Zap. Im. Akad. Nauk. 62, 1–24 (1889)

    Google Scholar 

  51. Markov, V.: Über die Funktionen, die in einem gegebenen Intervall möglichst wenig von Null abweichen. Math. Ann. 77, 213–258 (1916)

    Article  MathSciNet  MATH  Google Scholar 

  52. Milovanovic, G.V., Mitrinovic, D.S. Rassias, Th.M.: Topics in Polynomials, Extremal Problems, Inequalities, Zeros. World Scientific, Singapore (1994)

    Book  MATH  Google Scholar 

  53. Milówka, B.: Markovs inequality in Banach algebras. In: 5th Summer School in Potential Theory, Kraków (2006)

    Google Scholar 

  54. Mityagin, B.S.: Approximate dimension and bases in nuclear spaces. Russ. Math. Surv. 16, 59–127 (1961)

    Article  MATH  Google Scholar 

  55. 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)

    Article  MATH  Google Scholar 

  56. Nadzhmiddinov, D., Subbotin, Y.N.: Markov inequalities for polynomials on triangles. Mat. Zametki 46 (1989); English translation, Math. Notes 627–631 (1990)

    Google Scholar 

  57. Newman, P.D.J.: Derivative bounds for Müntz polynomials. J. Approx. Theory 18, 360–362 (1976)

    Article  MATH  Google Scholar 

  58. Pawlucki, W., Pleśniak, W.: Markov’s inequality and C functions on sets with polynomial cusps. Math. Ann. 275, 467–480 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  59. Pawlucki, W., Pleśniak, W.: Extension of C functions from sets with polynomial cusps. Studia Math. 88, 279–287 (1988)

    MathSciNet  MATH  Google Scholar 

  60. Pleśniak, W.: A Cantor regular set which does not have Markov’s property. Ann. Polon. Math. 51, 269–274 (1990)

    MathSciNet  MATH  Google Scholar 

  61. Pleśniak, W.: Recent progress in multivariate Markov inequality. Approximation Theory, In Memory of A. K. Varma, pp. 449–464. Marcel Dekker, New York (1998)

    Google Scholar 

  62. Pleśniak, W.: Markov’s inequality and the existence of an extension operator for C functions. J. Approx. Theory 61, 106–117 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  63. Pleśniak, W.: Extension and polynomial approximation of ultradifferentiable functions in R n. Bull. Soc. Roy. Sci. Liége 63(5), 393–402 (1994)

    MATH  Google Scholar 

  64. Pleśniak, W.: Inegalités de Markov en plusieurs variables. Int. J. Math. Math. Sci. 14, Article ID 24549, 1–12 (2006)

    Google Scholar 

  65. Pommerenke, Ch.: On the derivative of a polynomial. Michigan Math. J. 6, 373–375 (1959)

    Article  MathSciNet  MATH  Google Scholar 

  66. Sarantopoulos, Y.: Bounds on the derivatives of polynomials on Banach spaces. Math. Proc. Camb. Philos. Soc. 110, 307–312 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  67. Seeley, R.T.: Extension of C functions defined on a half-space. Proc. Am. Math. Soc. 15, 625–626 (1964)

    MathSciNet  MATH  Google Scholar 

  68. Siciak, J.: Degree of convergence of some sequences in the conformal mapping theory. Colloq. Math. 16, 49–59 (1967)

    MathSciNet  Google Scholar 

  69. Siciak, J.: Extremal plurisubharmonic functions in C n. Ann. Pol. Math. 39, 175–211 (1981)

    MathSciNet  MATH  Google Scholar 

  70. Siciak, J.: Highly non continuable functions on polynomially convex sets. Univ. Iagello Acta Math. 29, 95–107 (1985)

    MathSciNet  Google Scholar 

  71. Siciak, J.: Rapid polynomial approximation on compact sets in C n. Univ. Iagello. Acta Math. 30, 145–154 (1993)

    MathSciNet  Google Scholar 

  72. Stein, E.M.: Singular Integrals and Differentiability Properties of Functions. Princeton University Press, Princeton (1970)

    MATH  Google Scholar 

  73. 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)

    Chapter  Google Scholar 

  74. 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)

    Google Scholar 

  75. Toókos, F., Totik, V.: Markov inequality and Green functions. Rendiconti del Circolo Matematico di Palermo II 76, 91–102 (2005)

    Google Scholar 

  76. Totik, V.: Markov constants for Cantor sets. Acta Sci. Math. Szeged 60, 715–734 (1995)

    MathSciNet  MATH  Google Scholar 

  77. Wilhelmsen, D.R.: A Markov inequality in several dimensions. J. Approx. Theory 11, 216–220 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  78. Whitney, H.: Analytic extension of differentiable functions defined in closed sets. Trans. Am. Math. Soc. 36, 63–89 (1934)

    Article  MathSciNet  Google Scholar 

  79. 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)

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Nicholas J. Daras .

Editor information

Editors and Affiliations

Rights and permissions

Reprints and permissions

Copyright information

© 2014 Springer International Publishing Switzerland

About this chapter

Cite this chapter

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

Download citation

Publish with us

Policies and ethics