Sharp Constants of Approximation Theory. I. Multivariate Bernstein–Nikolskii Type Inequalities


Given a centrally symmetric convex body \(V\subset {{\mathbb {R}}}^m\), let \({{\mathcal {T}}}_{aV}\) be the set of all trigonometric polynomials with the spectrum in \(aV,\,a>0\), and let \(B_V\) be the set of all entire functions of exponential type with the spectrum in V. We discuss limit relations between the sharp constants in the multivariate Bernstein–Nikolskii inequalities defined by

$$\begin{aligned}&P_{p,q,D_N,a,V}:=a^{-N-m/p+m/q}\sup _{T\in {{\mathcal {T}}}_{aV}\setminus \{0\}}\frac{\Vert D_N(T)\Vert _{L_q\left( Q_{\pi }\right) }}{\Vert T\Vert _{L_p\left( Q_{\pi }\right) }},\\&E_{p,q,D_N,V}:= \sup _{f\in (B_V\cap L_p({{\mathbb {R}}}^m))\setminus \{0\}}\frac{\Vert D_N(f)\Vert _{L_q({{\mathbb {R}}}^m)}}{\Vert f\Vert _{L_p({{\mathbb {R}}}^m)}}, \end{aligned}$$

where \(0<p\le q\le {\infty },\,Q_{\pi }=\{x\in {{\mathbb {R}}}^m: \vert x_j\vert \le \pi ,\,1\le j\le m\},\) and \( D_N=\sum _{\vert {\alpha }\vert =N}b_{\alpha }D^{\alpha }\) is a differential operator with constant coefficients. We prove that

$$\begin{aligned} E_{p,q,D_N,V}\le \liminf _{a\rightarrow {\infty }}P_{p,q,D_N,a,V},\qquad E_{p,{\infty },D_N,V}= \lim _{a\rightarrow {\infty }}P_{p,{\infty },D_N,a,V}. \end{aligned}$$

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  1. 1.

    Akhiezer, N.I.: Lectures on the Theory of Approximation, 2nd edn. Nauka, Moscow (1965). (in Russian)

  2. 2.

    Arestov, V.V., On integral inequalities for trigonometric polynomials and their derivatives, Izv. Akad. Nauk SSSR, Ser. Mat. 45: 3–22 (in Russian). English transl. in Math. USSR-Izv. 18(1982), 1–17 (1981)

  3. 3.

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

  4. 4.

    Boas, R.P.: Entire Functions. Academic Press, New York (1954)

  5. 5.

    Dai, F., Gorbachev, D., Tikhonov, S.: Nikolskii constants for polynomials on the unit sphere, J. Anal. Math. (2019), in press; arXiv:1708.09837v1 (2017)

  6. 6.

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

  7. 7.

    Ganzburg, M.I.: The theorems of Jackson and Bernstein in \({\mathbb{R}}^m\), Uspekhi Mat. Nauk 34(1), 225–226 (1979). (in Russian); English transl. in Russian Math. Surveys 34(1), 221–222 (1979)

  8. 8.

    Ganzburg, M.I.: Bernstein-type, Multidimensional, inequalities, Ukrainskii Matematicheskii Zhurnal 34, 749–753 (in Russian). English transl. in Ukranian Math. J. 34(1982), 607–610 (1982)

  9. 9.

    Ganzburg, M.I.: Multidimensional polynomials of Levitan, in The Approximation of the Functions and Summation of the Series, pp. 112–117, Dniepropetrovsk State University, Dniepropetrovsk, (1990)

  10. 10.

    Ganzburg, M.I.: Polynomial inequalities on measurable sets and their applications. Constr. Approx. 17, 275–306 (2001)

  11. 11.

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

  12. 12.

    Gorbachev, D.V., Martyanov, I.A.: On the interrelation of the Nikol’skii constant for trigonometric polynomials and entire functions of exponential type. Chebyshevskii Sb. 19(2), 80–89 (2018). (in Russian)

  13. 13.

    Hörmander, L.: A new proof and a generalization of an inequality of Bohr. Math. Scand. 2, 33–45 (1954)

  14. 14.

    Hörmander, L.: Some inequalities for functions of exponential type. Math. Scand. 3, 21–27 (1955)

  15. 15.

    Kamzolov, A.I., On Riesz’s interpolation formula and Bernshtein’s inequality for functions on homogeneous spaces, Mat. Zametki 15, : 967–978 (in Russian). English transl. in Math. Notes 15(1974), 576–582 (1974)

  16. 16.

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

  17. 17.

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

  18. 18.

    Lewitan, B.M.: Über eine Verallgemeinerung der Ungleichungen von S. Bernstein und H. Bohr. Dokl. Akad. Nauk SSSR 15, 169–172 (1937)

  19. 19.

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

  20. 20.

    Nikolskii, S.M.: Inequalities for entire functions of finite degree and their application to the theory of differentiable functions of several variables, Trudy Mat. Inst. Steklov 38, 244–278 (1951). (in Russian); English transl. in Amer. Math. Soc. Transl. Ser. 2, 80, 1–38 (1969)

  21. 21.

    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-Verlag, New York-Heidelberg (1975)

  22. 22.

    Rahman, Q.I., Schmeisser, G.: \(L^p\) inequalities for entire functions of exponential type. Trans. Am. Math. Soc. 320, 91–103 (1990)

  23. 23.

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

  24. 24.

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

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Correspondence to Michael I. Ganzburg.

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Ganzburg, M.I. Sharp Constants of Approximation Theory. I. Multivariate Bernstein–Nikolskii Type Inequalities. J Fourier Anal Appl 26, 11 (2020).

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  • Sharp constants
  • Multivariate Bernstein–Nikolskii inequality
  • Trigonometric polynomials
  • Entire functions of exponential type
  • Multivariate Levitan’s polynomials

Mathematics Subject Classification

  • Primary 41A17
  • 41A63
  • Secondary 26D05
  • 26D10