Two sharp inequalities for operators in a Hilbert space

N.O. Kriachko

doi:10.15421/242206

Two sharp inequalities for operators in a Hilbert space

N.O. Kriachko (Oles Honchar Dnipro National University)

Abstract

In this paper we obtained generalisations of the L. V. Taikov’s and N. Ainulloev’s sharp inequalities, which estimate a norm of function's first-order derivative (L. V. Taikov) and a norm of function's second-order derivative (N. Ainulloev) via the modulus of continuity or the modulus of smoothness of the function itself and the modulus of continuity or the modulus of smoothness of the function's second-order derivative. The generalisations are obtained on the power of unbounded self-adjoint operators which act in a Hilbert space. The moduli of continuity or smoothness are defined by a strongly continuous group of unitary operators.

Keywords

Hilbert space; self-adjoint operator; modulus of smoothness; partition of unity; strongly continuous group of unitary operators

MSC 2020

41A17; 41A44; 26E10

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References

Ainulloev N. "An exact estimate of the second derivative in $$$L_p$$$", Math. Notes AN SSSR, 1991; 49(5): pp. 443-445. doi:10.1007/BF01142636

Babenko V.F., Bilichenko R.O. "Approximation of unbounded operators by bounded operators in a Hilbert space", Ukrainian Math. J., 2009; 61(2): pp. 179-187. doi:10.1007/s11253-009-0212-2

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

Taikov L.V. "Some exact inequalities in approximation theory for functions", Analys. Math., 1976; 2: pp. 77-85.

Akhiezer N.I., Glazman I.M. Theory of linear operators in Hilbert space, Vyshcha shkola, 1977.

Babenko V.F., Bilichenko R.O. "The best approximation of classes, defined by powers of self-adjoint operators acting in Hilbert space, by other classes", Res. Math., 2009; 17: pp. 23-30.

Kuptsov N.P. "Direct and converse theorems of approximation theory and semigroups of operators", russ. Math. Surv., 1968; 23(4): pp. 115-177.

Trenogin V.A. Functional Analysis, Nauka, 1980.