Extremal Nikolskii – Bernstein- and Turán-type problems for Dunkl transform
https://doi.org/10.22405/2226-8383-2019-20-3-394-400
Abstract
We study the interrelation between the extremal Turán-type problems and Nikolskii – Bernstein problems for nonnegative functions on Rd with the Dunkl weight. The Turán problem is to find the supremum of a given moment of a positive definite (with respect to the Dunkl transform) function with a support in the Euclidean ball and a fixed value at zero. In the sharp L1-Nikolskii–Bernstein inequality, the supremum norm of the Dankl Laplacian of an entire function of exponential spherical type with the unit L1-norm is estimated. Extremal Feuér and Beaumann problems is also mentioned. The Dunkl transform covers the case of the classical Fourier transform in the case of unit weight.
Nikolskii–Bernstein inequalities are classical in approximation theory, and the Turán-type problems have applications in metric geometry. Nevertheless, we prove that they have the same answer, which is given explicitly. The easy proof is relied on our old results from the theory of solving extremal problems to the Dunkl transform.
About the Authors
Dmitry Viktorovich GorbachevRussian Federation
Nikolai Nikolaevich Dobrovol’sky
Russian Federation
References
1. Arestov V. V., Berdysheva E. E. The Turán problem for a class of polytopes // East J. Approx. 2002. Vol. 8, no. 2. P. 381–388.
2. Bianchi G., Kelly M. A Fourier analytic proof of the Blaschke–Santal’o inequality // Proc. Amer. Math. Soc. 2015. Vol. 143. P. 4901–4912.
3. Dai F., Gorbachev D., Tikhonov S. Nikolskii constants for polynomials on the unit sphere // arXiv:1708.09837. 2017; J. d’Analyse Math. 2019 (to appear).
4. Gorbachev D. V. An extremal problem for periodic functions with supports in the ball // Math. Notes. 2001. Vol. 69, no. 3–4. P. 313–319.
5. Gorbachev D. V. Selected problems of function theory and approximation theory, and their applications. Tula: Grif and Co., 2005. (In Russ.)
6. Gorbachev D. V., Ivanov V. I., Ofitserov E. P., Smirnov O. I. Some extremal problems of harmonic analysis and approximation theory // Chebyshevskii Sbornik. 2017. Vol. 18, no. 4. P. 139–166. (In Russ.)
7. Gorbachev D. V., Ivanov V. I. Bohman extremal problem for the Dunkl transform // Proc. Steklov Inst. Math. 2017. Vol. 297 (Suppl 1). P. 88–96.
8. Gorbachev D. V., Tikhonov S. Y. Wiener’s problem for positive definite functions // Math. Zeit. 2018. Vol. 289, no. 3–4. P. 859-874.
9. Gorbachev D. V., Dobrovolskii N. N. Nikolskii constants in Lp(R, |x|2α+1dx) spaces // Chebyshevskii Sbornik. 2018. Vol. 19, no. 2. P. 67–79. (In Russ.)
10. Gorbachev D. V. Nikolskii – Bernstein constants for nonnegative entire functions of exponential type on the axis // Trudy Instituta Matematiki i Mekhaniki UrO RAN. 2018. Vol. 24, no. 4. P. 92–103. (In Russ.)
11. Gorbachev D. V., Ivanov V. I. Nikol’skii – Bernstein constants for entire functions of exponential spherical type in weighted spaces // Trudy Instituta Matematiki i Mekhaniki URO RAN. 2019. Vol. 25, no. 2. P. 75–87. (In Russ.)
12. Gorbachev D. V., Ivanov V. I. Tur’an, Fej’er and Bohman extremal problems for the multivariate Fourier transform in terms of the eigenfunctions of a Sturm–Liouville problem // Sb. Math. 2019. Vol. 210, no. 6. P. 809–835.
13. Ivanov A. V. Some extremal problem for entire functions in weighted spaces // Izv. Tul. Gos. Univ., Ser. Estestv. Nauki. 2010. No. 1. P. 26–44. (In Russ.)
14. Kolountzakis M. N., R’ev’esz Sz.Gy. On a problem of Turán about positive definite functions // Proc. Amer. Math. Soc. 2003. Vol. 131. P. 3423–3430.
15. Rösler M. Dunkl Operators: Theory and Applications. Lecture Notes in Math. Vol. 1817. Berlin: Springer, 2003. P. 93–135.
16. Siegel C. L. Über Gitterpunkte in konvexen Körpern und damit zusammenhängendes Extremal problem // Acta Math. 1935. Vol. 65. P. 307–323.
17. Vaaler J. D. Some extremal functions in Fourier analysis // Bull. Amer. Math. Soc. (New Series). 1985. Vol. 12, no. 2. P. 183–216.
Review
For citations:
Gorbachev D.V., Dobrovol’sky N.N. Extremal Nikolskii – Bernstein- and Turán-type problems for Dunkl transform. Chebyshevskii Sbornik. 2019;20(3):394-400. (In Russ.) https://doi.org/10.22405/2226-8383-2019-20-3-394-400
Views:
350
Email this article 