Approximation of unbounded functionals by the bounded ones in Hilbert space

V.F. Babenko; R.O. Bilichenko

doi:10.15421/241201

Approximation of unbounded functionals by the bounded ones in Hilbert space

V.F. Babenko (Oles Honchar Dnipropetrovsk National University), https://orcid.org/0000-0001-6677-1914
R.O. Bilichenko (Oles Honchar Dnipropetrovsk National University)

Abstract

We obtained the value of the best approximation of unbounded functional $$$F_f(x) = (A^kx, f)$$$ on the class $$$\{ x\in D(A^r) \colon \| A^r x \| \leqslant 1 \}$$$ by linear bounded functionals ($$$A$$$ is a self-adjoint operator in the Hilbert space $$$H$$$, $$$f\in H$$$, $$$k < r$$$).

Keywords

functional; self-adjoint operator; Hilbert space

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References

Arestov V.V., Gabushin V.N. "The best approximation of unbounded operators by the bounded ones", Izv. vuzov. Matematika, 1995; 11: pp. 42-63. (in Russian)

Arestov V.V. "Approximation of unbounded operators by the bounded ones and related extremum problems", Uspekhi mat. nauk, 1996; 51(6): pp. 88-124. (in Russian) doi:10.1070/RM1996v051n06ABEH003001

Akhiezer N.I., Glazman I.M. Theory of linear operators in Hilbert space, Moscow, 1966; 544 p. (in Russian)

Babenko V.F., Bilichenko R.O. "Inequalities of Taikov type for self-adjoint operators in Hilbert space", Trudy IPMM, 2010; 21: pp. 11-18. (in Russian)

Berezanskij Yu.M., Us G.F., Sheftel' Z.G. Functional analysis, Kyiv, 1990; 600 p. (in Russian)

Korneichuk N.P., Babenko V.F., Kofanov V.A., Pichugov S.A. Inequalities for derivatives and their applications, Nauk. dumka, Kyiv, 2003; 590 p. (in Russian)

Stechkin S.B. "Inequalities between norms of derivatives of arbitrary function", Acta scient. math., 1965; 26: pp. 225-230. (in Russian)

Stechkin S.B. "The best approximation of linear operators", Matem. zametki, 1967; 1(2): pp. 137-148. (in Russian) doi:10.1007/BF01268056

Taikov L.V. "Inequalities of Kolmogorov type and the best formulas of numeric differentiation", Matem. zametki, 1968; 4(2): pp. 223-238. (in Russian) doi:10.1007/BF01094964

Shadrin A.Yu. "Inequalities of Kolmogorov type and estimates of spline-interpolation for periodic classes $$$W^m_2$$$", Matem. zametki, 1990; 48(4): pp. 132-139. (in Russian) doi:10.1007/BF01139609