Skip to main content

Advertisement

SpringerLink
Log in

Sharp Constants of Approximation Theory: VI. Weighted Polynomial Inequalities of Different Metrics on the Multidimensional Cube and Ball

  • Published:
Constructive Approximation Aims and scope

Abstract

We prove limit equalities between the sharp constants in weighted Nikolskii-type inequalities for multivariate polynomials on an m-dimensional cube and ball and the corresponding constants for entire functions of exponential type.

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

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Arestov, V.V., Deikalova, M.V.: Nikol’skii inequality for algebraic polynomials on a multidimensional Euclidean sphere. Trudy IMM UrO RAN 19(2), 34–47 (2013) (in Russian); English transl. in Proc. Steklov Inst. Math. 284(Suppl. 1), S9–S23 (2014)

  2. Arestov, V., Deikalova, M.: Nikol’skii inequality between the uniform norm and \(L_q\)-norm with ultraspherical weight of algebraic polynomials on an interval. Comput. Methods Funct. Theory 15, 689–708 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  3. Arnold, V.I., Varchenko, A.N., Givental, A.B., Khovanskii, A.G.: Singularities of functions, wave fronts, caustics and multidimensional integrals. Translated from the Russian. Soviet Sci. Rev. Sect. C Math. Phys. Rev. 4, Mathematical physics reviews, Vol. 4, pp. 1–92, Harwood Academic Publ., Chur (1984)

  4. Bernstein, S.N.: On entire functions of finite degree of several variables. Dokl. Akad. Nauk SSSR 60, 949–952 (1948). (in Russian)

    MathSciNet  Google Scholar 

  5. Berens, H., Butzer, P.L., Pawelke, S.: Limitierungsverfahren von Reihen mehrdimensionaler Kugelfunktionen und deren Saturationsverhalten. Publ. Res. Inst. Math. Sci. (Kyoto) Ser. A 4, 201–268 (1968/1969)

  6. Butzer, P.L., Stens, R.L., Wehrens, M.: Higher order moduli of continuity based on the Jacobi translation operator and best approximation. C. R. Math. Rep. Acad. Sci. Can. 11(2), 83–88 (1980)

    MathSciNet  MATH  Google Scholar 

  7. Dai, F., Gorbachev, D., Tikhonov, S.: Nikolskii constants for polynomials on the unit sphere. J. d’Analyse Math. 140, 161–185 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  8. Daugavet, I.K.: On Markov–Nikolskii type inequalities for algebraic polynomials in the multidimensional case. Dokl. Akad. Nauk SSSR 207, 521–522 (1972) (in Russian). English transl. in Soviet Math. Dokl. 13, 1548–1550 (1972)

  9. Daugavet, I.K.: Some inequalities for polynomials in the multidimensional case. Numer. Methods (Leningr. Univ.) 10, 3–26 (1976). (in Russian)

    MathSciNet  Google Scholar 

  10. Deikalova, M., Rogozina, V.V.: Jackson–Nikol’skii inequality between the uniform and integral norms of algebraic polynomials on a Euclidean sphere. Trudy IMM UrO RAN 18(4), 162–171 (2012). (in Russian)

    Google Scholar 

  11. Ditzian, Z., Prymak, A.: On Nikolskii inequalities for Lorentz spaces. Rocky Mt. J. Math. 40(1), 209–223 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  12. Ditzian, Z., Prymak, A.: Nikolskii inequalities for domains in \(\mathbb{R}^{d}\). Constr. Approx. 44(1), 23–51 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  13. Dunford, N., Schwartz, J.T.: Linear Operators, Part I: General Theory. Wiley, New York (1988)

    MATH  Google Scholar 

  14. Ganzburg, M.I.: An exact inequality for the increasing rearrangement of a polynomial in \(m\) variables. Teor. Funktsiǐ Funktsional. Anal. i Prilozhen. (Kharkov) 31, 16–24 (1978). (in Russian)

    Google Scholar 

  15. Ganzburg, M.I.: A direct and inverse theorem on approximation by polynomials on the \(m\)-dimensional ball. Trudy Mat. Inst. Steklov 180, 91–92 (1987) (in Russian); English transl. in Proc. Steklov Inst. Math. 180(3), 103–105 (1989)

  16. Ganzburg, M.I.: Polynomial approximation on the \(m\)-dimensional ball. In: Chui, C.K., Schumaker, L.L. (eds.) Approximation Theory IX. Theoretical Aspects, vol. 1, pp. 141–148. Vanderbilt University Press, Nashville (1998)

    Google Scholar 

  17. Ganzburg, M.I.: Invariance theorems in approximation theory and their applications. Constr. Approx. 27, 289–321 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  18. Ganzburg, M.I.: Multivariate polynomial inequalities of different \(L_{p, W}(V)\)-metrics with \(k\)-concave weights. Acta Math. Hungar. 150(1), 99–120 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ganzburg, M.I.: Sharp constants in V. A. Markov–Bernstein type inequalities of different metrics. J. Approx. Theory 215, 92–105 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  20. Ganzburg, M.I.: Sharp constants of approximation theory. II. Invariance theorems and certain multivariate inequalities of different metrics. Constr. Approx. 50, 543–577 (2019)

    Article  MathSciNet  MATH  Google Scholar 

  21. Ganzburg, M.I.: Sharp constants of approximation theory. I. Multivariate Bernstein–Nikolskii type inequalities. J. Fourier Anal. Appl. 26(1), 11 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ganzburg, M.I.: Sharp constants of approximation theory. III. Certain polynomial inequalities of different metrics on convex sets. J. Approx. Theory 252, 105351 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  23. Ganzburg, M.I.: Sharp constants of approximation theory. V. An asymptotic equality related to polynomials with given Newton polyhedra. J. Math. Anal. Appl. 499, 125026 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  24. Ganzburg, M.I.: Sharp constants of approximation theory. IV. Asymptotic relations in general settings. Anal. Math. accepted for publication. arXiv:2002.10512

  25. Ganzburg, M.I., Tikhonov, S.Yu.: On sharp constants in Bernstein–Nikolskii inequalities. Constr. Approx. 45, 449–466 (2017)

  26. Geronimus, Y.L.: On an extremal problem of Chebyshev. Izv. Akad. Nauk SSSR Ser. Mat. 2(4), 445–456 (1938). (in Russian)

    MATH  Google Scholar 

  27. Gorbachev, D.V., Martyanov, I.A.: Bounds of the Nikolskii polynomial constants in \(L^p\) with a Gegenbauer weight. Trudy Inst. Mat. Mekh. UrO RAN 26(4), 126–137 (2020). (in Russian)

    Google Scholar 

  28. Kacnelson, V.E.: Equivalent norms in spaces of entire functions, Mat. Sb. 92(134)(1), 34–54 (1973) (in Russian). English translation in Math. USSR-Sb. 21(1), 33–55 (1973)

  29. Kroó, A., Schmidt, D.: Some extremal problems for multivariate polynomials on convex bodies. J. Approx. Theory 90, 415–434 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  30. Labelle, G.: Concerning polynomials on the unit interval. Proc. Am. Math. Soc. 20, 321–326 (1969)

    Article  MathSciNet  MATH  Google Scholar 

  31. Levin, E., Lubinsky, D.: \(L_p\) Christoffel functions, \(L_p\) universality, and Paley–Wiener spaces. J. D’Analyse Math. 125, 243–283 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  32. Levin, E., Lubinsky, D.: Asymptotic behavior of Nikolskii constants for polynomials on the unit circle. Comput. Methods Funct. Theory 15, 459–468 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  33. Lizorkin, P.I., Nikol’skii, S.M.: A theorem concerning approximation on the sphere. Anal. Math. 9, 207–221 (1983)

    Article  MathSciNet  MATH  Google Scholar 

  34. Lupas, A.: An inequality for polynomials. Univ. Beograd. Publ. Elektrotehn. Fak. Ser. Mat. Fiz. 461–497, 241–243 (1974)

    MathSciNet  MATH  Google Scholar 

  35. Migliorati, G.: Multivariate Markov-type and Nikolskii-type inequalities for polynomials associated with downward closed multi-index sets. J. Approx. Theory 189, 137–159 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  36. Milovanović, G.V., Mitrinović, D.S., Rassias, Th.M.: Topics in Polynomials: Extremal Problems, Inequalities, Zeros. World Scientific, Singapore (1994)

  37. Nessel, R.J., Wilmes, G.: Nikolskii-type inequalities for trigonometric polynomials and entire functions of exponential type. J. Austral. Math. Soc. Ser. A 25, 7–18 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  38. Nikolskii, S.M.: Approximation of Functions of Several Variables and Imbedding Theorems, Nauka, Moscow, 1969 (in Russian); English edition: Die Grundlehren der Mathematischen Wissenschaften, vol. 205. Springer, New York (1975)

    Google Scholar 

  39. Polya, G., Szegö, G.: Problems and Theorems in Analysis II. Springer, Berlin (1998)

    Book  MATH  Google Scholar 

  40. Rockafellar, R.T.: Convex Analysis. Princeton University Press, Princeton (1970)

    Book  MATH  Google Scholar 

  41. Rudin, W.: Functional Analysis. McGraw-Hill, New York (1973)

    MATH  Google Scholar 

  42. Rustamov, K.P.: On equivalence of different moduli of smoothness on the sphere. Trudy Mat. Inst. Steklov 204, 274–304 (1993) (in Russian). English transl. in Proc. Steklov Inst. Math. 204(3), 235–260 (1994)

  43. Simonov, I.E., Glazyrina, P.Y.: Sharp Markov–Nikolskii inequality with respect to the uniform norm and the integral norm with Chebyshev weight. J. Approx. Theory 192, 69–81 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  44. Stein, E., Weiss, G.: Introduction to Fourier Analysis on Euclidean Spaces. Princeton University Press, Princeton (1971)

    MATH  Google Scholar 

  45. Timan, A.F.: Theory of Approximation of Functions of a Real Variable. Pergamon Press, New York (1963)

    MATH  Google Scholar 

  46. Watson, G.N.: A Treatise on the Theory of Bessel Functions, 2nd edn. Cambridge University Press, Cambridge (1944) (reprinted in 1952, 1958, 1962, 1966)

  47. Wherens, M.: Best approximation on the unit sphere in \(\mathbb{R}^{k}\). In: Functional Analysis and Approximation (Oberwolfach, 1980), pp. 233–245, Internat. Ser. Numer. Math., 60. Birkhauser, Basel (1981)

  48. Xu, Y.: Approximation by means of h-harmonic polynomials on the unit sphere. Adv. Comput. Math. 21, 37–58 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  49. Xu, Y.: Weighted approximation of functions on the unit sphere. Constr. Approx. 21, 1–28 (2005)

    MathSciNet  MATH  Google Scholar 

  50. Xu, Y.: Generalized translation operator and approximation in several variables. J. Comput. Appl. Math. 178, 489–512 (2005)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We are grateful to both anonymous referees for valuable suggestions.

Author information

Authors and Affiliations

  1. Hampton, USA

    Michael I. Ganzburg

Authors
  1. Michael I. Ganzburg
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Michael I. Ganzburg.

Additional information

Communicated by Yuan Xu.

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Ganzburg, M.I. Sharp Constants of Approximation Theory: VI. Weighted Polynomial Inequalities of Different Metrics on the Multidimensional Cube and Ball. Constr Approx 56, 649–685 (2022). https://doi.org/10.1007/s00365-022-09587-0

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00365-022-09587-0

Keywords

Mathematics Subject Classification

Search

Navigation