The Taikov type inequalities in rigged Hilbert spaces

V. Babenko; Yu. Babenko; N. Kriachko; D. Skorokhodov

doi:10.15421/242510

The Taikov type inequalities in rigged Hilbert spaces

V. Babenko (Oles Honchar Dnipro National University), https://orcid.org/0000-0001-6677-1914
Yu. Babenko (Kennesaw State University)
N. Kriachko (Oles Honchar Dnipro National University)
D. Skorokhodov (Oles Honchar Dnipro National University), https://orcid.org/0000-0001-8494-5885

Abstract

For the chain $$$H_+\subset H_0\subset H_-$$$ of rigged Hilbert spaces we solve the problem of finding sharp Taikov type inequalities that estimate the value of generalized element $$$\alpha\in H_-$$$ on smooth elements $$$u\in H_+$$$ in terms of the norm of the element $$$u$$$ in $$$H_+$$$. Also, we solve the problem of approximating generalized elements $$$\alpha\in H_-$$$ by regular elements $$$x\in H_0$$$ on the class $$$W_+ = \{u\in H_+\,:\,\|u\|_{H_+}\le 1\}$$$. We consider some applications of these results, in particular, to finding sharp Taikov-type inequalities for partial derivatives of multi-variate periodic functions.

Keywords

Kolmogorov type inequalities; rigged Hilbert spaces; Taikov type inequalities; the Stechkin problem

MSC 2020

Pri 47A63, 41A35, 26D10, Sec 41A65

Full Text:

PDF

References

Babenko V., Babenko Yu, Kriachko N. "A solution to Stechkin's problem for function of a self-adjoint operator in a Hilbert space, Taikov type inequalities, and their applications", Jaen J. Approx., 2022; 13: pp. 1-28.

Babenko V., Babenko Yu., Kriachko N., Skorokhodov D. "On Hardy-Littlewood-Polya and Taikov Type Inequalities For Multiple Operators in Hilbert Spaces", Analysis Math., 2021; 47(4): pp. 709-745. doi:10.1007/s10476-021-0104-8

Babenko V.F., Bilichenko R.O. "Taikov type inequalities for self-adjoint operators in Hilbert space", Trudi IPMM NAN Ukraine, 2010; 21: pp. 1-7.

Berezanskij Yu.M., Us G.F., Sheftel Z.G. Functional Analysis, Elsevier Sci., 2003.

Buslaev A.P., Tikhomirov V.M. "Inequalities for derivatives in the multidimensional case", Math. Notes, 1979; 25(1): pp. 32-40. doi:10.1007/BF01138146

Gabushin V.N. "The best approximation of the differentiation operator on the half line", Math. Notes, 1969; 6(5): pp. 804-810. doi:10.1007/BF01101408

Kalyabin G.A. "Sharp Constants in Inequalities for Intermediate Derivatives (the Gabushin Case)", Func. Anal. Appl., 2004; 38(3): pp. 184-191. doi:10.1023/B:FAIA.0000042803.72039.20

Lunyov A.A. "Sharp constants in inequalities for intermediate derivatives", Ukr. Math. Bull., 2007; 4(3): pp. 421-433.

Lunyov A.A. "Exact constants in inequalities for intermediate derivatives in an $$$n$$$-dimensional space", Math. Notes, 2009; 85(3): pp. 458-462. doi:10.1134/S0001434609030171

Lunev A.A., Oridoroga L.L. "Exact constants in generalized inequalities for intermediate derivatives", Math. Notes, 2009; 85(5): pp. 703-711. doi:10.1134/S0001434609050101

Shadrin A.Yu. "Inequalities of Kolmogorov type and estimates of spline-interpolation for periodic classes $$$W^m_2$$$", Math. Notes, 1990; 48(4): pp. 1058--1063. doi:10.1007/BF01139609

Skorokhodov D. "The Landau–Kolmogorov problem on a finite interval in the Taikov case", J. Approx. Theory, 2022; 280(8): 105771. doi:10.1016/j.jat.2022.105771

Stechkin S.B. "Best approximation of linear operators", Math. Notes, 1967; 1(2): pp. 91-99. doi:10.1007/BF01268056

Taikov L.V. "Kolmogorov-type inequalities and the best formulas for numerical differentiation", Math. Notes, 1968; 4(2): pp. 631-634. doi:10.1007/BF01094964

Yosida K. Functional Analysis, Springer, 1995. doi:10.1007/978-3-642-61859-8

Babenko V., Kriachko N. "On Hardy-Littlewood-Pólya type inequalities for operators in Hilbert space", Res. Math., 2014; 22: pp. 3-7.

Babenko V.F., Bilichenko R.O. "Inequality of Taikov type for powers of normal operators in Hilbert space", Res. Math., 2011; 19: pp. 3-7.

Babenko V.F., Korneichuk N.P., Kofanov V.A., Pichugov S.A. Inequalities for derivatives and their applications, Naukova dumka, 2003.

Ilyin A.A. "Best constants in a class of polymultiplicative inequalities for derivatives", Sb. Math., 1998; 189(9): pp. 1335-1359.