D.V. Gorbachev. Nikolskii – Bernstein constants for nonnegative entire functions of exponential type on the axis ... P. 92-103

We investigate a weighted version of the Nikolskii-Bernstein inequality
\[ \|\Lambda_{\alpha}^{k}f\|_{q,\alpha}\le \mathcal{L}(\alpha,p,q,k)\sigma^{(2\alpha+2)(1/p-1/q)+k}\|f\|_{p,\alpha},\quad \alpha\ge -1/2, \]
on the subspace $\mathcal{E}^{\sigma}\cap L^{p}(\mathbb{R},|x|^{2\alpha+1}\,dx)$ of entire functions of exponential type. Here $\Lambda_{\alpha}$ is the Dunkl differential-difference operator whose second power generates  the Bessel differential-difference operator $B_{\alpha}$. For $(p,q)=(1,\infty)$, we compute the following sharp constants for nonnegative functions:
\[ \mathcal{L}_{0}^{*}(\alpha)_{+}=\frac{1}{2^{2\alpha+2}},\quad \mathcal{L}_{1}^{*}(\alpha)_{+}=\frac{1}{2^{2\alpha+4}(\alpha+2)}, \]
where $\mathcal{L}_{r}^{*}(\alpha)_{+}= (\alpha+1)c_{\alpha}^{-2}\mathcal{L}(\alpha,1,\infty,2r)_{+}$ denotes the normalized Nikolskii-Bernstein constant. There are unique (up to a constant factor) extremizers $j_{\alpha+1}^{2}(x/2)$ and $x^{2}j_{\alpha+2}^{2}(x/2)$, respectively. These results are proved with the use of the Markov quadrature formula with nodes at zeros of the Bessel function and the following generalization of Arestov, Babenko, Deikalova, and Horv$\acute{\mathrm{a}}$th's recent result:
\[ \mathcal{L}(\alpha,p,\infty,2r)=\sup B_{\alpha}^{r}f(0),\quad r\in \mathbb{Z}_{+}, \]
where the supremum is taken over all even real functions on $\mathbb{R}$ belonging to $\mathcal{E}_{p,\alpha}^{1}$. Our approach is based on the one-dimensional Dunkl harmonic analysis. In particular, we use the even positive Dunkl-type generalized translation operator $T_{\alpha}^{t}$, which is bounded on $L^{p}(\mathbb{R},|t|^{2\alpha+1}\,dt)$ with constant 1, is invariant on the subspace $\mathcal{E}_{p,\alpha}^{\sigma}$, and commutes with $B_{\alpha}$.

Keywords: weighted Nikolskii-Bernstein inequality, sharp constant, entire function of exponential type, Dunkl transform, generalized translation operator, Bessel function

Received September 05, 2018

Revised November 15, 2018

Accepted November 19, 2018

Funding Agency: This work was supported by the Russian Science Foundation (project no. 18-11-00199).

Dmitry Viktorovich Gorbachev, Dr. Phys.-Math. Sci., Prof., Tula State University, Tula, 300012 Russia, e-mail: dvgmail@mail.ru


1.   Achieser N.I. Theory of approximation. N Y: Dover, 2004, 307 p. ISBN: 0486495434 .

2.   Andersen N.B., de Jeu M. Elementary proofs of Paley–Wiener theorems for the Dunkl transform on the real line. Int. Math. Res. Notices, 2005, vol. 2005, no. 30, pp. 1817–1831.

3.   Arestov V.V. Inequality of different metrics for trigonometric polynomials. Math. Notes, 1980, vol. 27, no. 4, pp. 265–269. doi: 10.1007/BF01140526

4.   Arestov V., Babenko A., Deikalova M., Horv$\acute{\mathrm{a}}$th $\acute{\mathrm{A}}$. Nikol’skii inequality between the uniform norm and integral norm with Bessel weight for entire functions of exponential type on the half-line. Anal. Math., 2018, vol. 44, no. 1, pp. 21–42. doi: 10.1007/s10476-018-0103-6

5.   Bateman G., Erd$\acute{\mathrm{e}}$lyi A., et al. Higher transcendental functions. Vol. II. N Y: McGraw Hill Book Company, 1953, 396 p. ISBN: 0486446158 .

6.   Dai F., Gorbachev D., Tikhonov S. Nikol’skii constants for polynomials on the unit sphere [e-resource]. 21 p. Available at: https://arxiv.org/pdf/1708.09837.pdf

7.   Frappier C., Olivier P. A quadrature formula involving zeros of Bessel functions. Math. Comp., 1993, vol. 60, pp. 303–316. doi: 10.1090/S0025-5718-1993-1149290-5

8.   Ghanem R.B., Frappier C. Explicit quadrature formulae for entire functions of exponential type. J. Approx. Theory, 1998, vol. 92, no. 2, pp. 267–279. doi: 10.1006/jath.1997.3122

9.   Gorbachev D.V. Extremum problems for entire functions of exponential spherical type. Math. Notes, 2000, vol. 68, no. 2, pp. 159–166. doi: 10.1007/BF02675341

10.   Gorbachev D.V. Extremal problem for periodic functions supported in a ball. Math. Notes, 2001, vol. 69, no. 3-4, pp. 313–319. doi: 10.1023/A:1010275206760

11.   Gorbachev D.V. An integral problem of Konyagin and the (C,L)-constants of Nikol’skii. Proc. Steklov Inst. Math., 2005, suppl. 2, pp. S117–S138.

12.   Gorbachev D.V., Dobrovol’skii N.N. The Nikol’skii constants in spaces $L^{p}(\mathbb{R},|x|^{2\alpha+1}\,dx)$. Chebyshevskii Sb., 2018, no. 2, pp. 67–79.

13.   Gorbachev D.V., Ivanov V.I. Gauss and Markov quadrature formulae with nodes at zeros of eigenfunctions of a Sturm–Liouville problem, which are exact for entire functions of exponential type. Sbornik: Math., 2015, vol. 206, no. 8, pp. 1087–1122. doi: 10.1070/SM2015v206n08ABEH004490

14.   Gorbachev D.V., Ivanov V.I. Approximation in $L_2$ by partial integrals of the Fourier transform over the eigenfunctions of the Sturm – Liouville operator. Math. Notes, 2016, vol. 100, no. 3–4, pp. 540–549. doi: 10.1134/S000143461609025X

15.    Gorbachev D.V., Ivanov V.I., Tikhonov S.Y. Positive $L^p$-bounded Dunkl-type generalized translation operator and its applications. Constr. Approx., 2018, pp. 1–51. doi: 10.1007/s00365-018-9435-5

16.   Grozev G.R., Rahman Q.I. A quadrature formula with zeros of Bessel functions as nodes. Math. Comp., 1995, vol. 64, pp. 715–725. doi: 10.1090/S0025-5718-1995-1277767-2

17.   Levin B.Ya. Lectures on entire functions. English revised edition. Providence, RI: Amer. Math. Soc., 1996, 248 p. ISBN: 0-8218-0282-8 . Original Russian text published in Levin B.Ya. Tselye funktsii. Moscow: Mos. Gos. Univ., Mekh.-Mat. Fak., 1971, 124 p.

18.   Rahman Q.I., Schmeisser G. $L^p$ inequalities for entire functions of exponential type. Trans. Amer. Math. Soc., 1990, vol. 320, no. 1, pp. 91–103. doi: 10.1090/S0002-9947-1990-0974526-4

19.   R$\ddot{\mathrm{o}}$sler M. Dunkl Operators: Theory and Applications. In: Koelink E., Van Assche W. (eds) Lecture Notes in Math., vol. 1817. Berlin: Springer, 2003, pp. 93–135. doi: 10.1007/3-540-44945-0_3

20.   Nikol’skii S.M. Approximation of functions of several variables and imbedding theorems. Berlin; Heidelberg; N Y: Springer, 1975, 420 p. doi: 10.1007/978-3-642-65711-5 . Original Russian text published in Nikol’skii S.M. Priblizhenie funktsii mnogikh peremennykh i teoremy vlozheniya. Moscow: Nauka Publ., 1977, 480 p.

21.   Platonov S.S. Bessel harmonic analysis and approximation of functions on the half-line. Izvestiya: Math., 2007, vol. 71, no. 5, pp. 1001–1048. doi: 10.1070/IM2007v071n05ABEH002379

22.   R$\acute{\mathrm{e}}$v$\acute{\mathrm{e}}$sz Sz.Gy. Tur$\acute{\mathrm{a}}$n’s extremal problem on locally compact abelian groups. Anal. Math., 2011, vol. 37, no. 1, pp. 15–50. doi: 10.1007/s10476-011-0102-3 .

23.   Soltani F. $L^p$-Fourier multipliers for the Dunkl operator on the real line. J. Funct. Anal., 2004, vol. 209, no. 1, pp. 16–35. doi: 10.1016/j.jfa.2003.11.009 .

24.   Timan A.F. Theory of Approximation of Functions of a Real Variable. N Y: Pergamon Press, 1963, 631 p.