Abstract
For an arbitrary self-adjoint operator B in a Hilbert space \(\mathfrak{H}\), we present direct and inverse theorems establishing the relationship between the degree of smoothness of a vector \(x \in \mathfrak{H}\) with respect to the operator B, the rate of convergence to zero of its best approximation by exponential-type entire vectors of the operator B, and the k-modulus of continuity of the vector x with respect to the operator B. The results are used for finding a priori estimates for the Ritz approximate solutions of operator equations in a Hilbert space.
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REFERENCES
M. L. Gorbachuk, “On analytic solutions of differential-operator equations,” Ukr. Mat. Zh., 52, No.5, 596–607 (2000).
N. P. Kuptsov, “Direct and inverse theorems of approximation theory and semigroups of operators,” Usp. Mat. Nauk., 23, Issue 4, 118–178 (1968).
N. I. Akhiezer and I. M. Glazman, Theory of Linear Operators in a Hilbert Space [in Russian], Nauka, Moscow (1966).
N. I. Akhiezer, Lectures on Relativity Theory [in Russian], Nauka, Moscow (1965).
M. L. Gorbachuk and V. I. Gorbachuk, “Spaces of infinitely differentiable vectors of a closed operator and their application to problems of approximation,” Usp. Mat. Nauk, 48, Issue 4, 180 (1993).
V. I. Gorbachuk and M. L. Gorbachuk, “Operator approach to problems of approximation,” Algebra Analiz, 9, Issue 6, 90–108 (1997).
A. I. Stepanets and A. S. Serdyuk, “Direct and inverse theorems in the theory of approximation of functions in the space S p,” Ukr. Mat. Zh., 54, No.1, 106–124 (2002).
N. I. Chernykh, “On Jackson inequalities in L 2,” Tr. Mat. Inst. Akad. Nauk SSSR, 88, 71–74 (1967).
S. G. Mikhlin, Variational Methods in Mathematical Physics [in Russian], Nauka, Moscow (1970).
A. Yu. Luchka and G. F. Luchka, Appearance and Development of Direct Methods in Mathematical Physics [in Russian], Naukova Dumka, Kiev (1970).
A. V. Dzhishkariani, “On the rate of convergence of the Ritz approximation method,” Zh. Vychisl. Mat. Mat. Fiz., 3, No.4, 654–663 (1963).
M. Sh. Birman and M. Z. Solom'yak, Spectral Theory of Self-Adjoint Operators in a Hilbert Space [in Russian], Leningrad University, Leningrad (1980).
Ya. V. Radyno, “Spaces of vectors of exponential type,” Dokl. Akad. Nauk Bel. SSR, 27, No.9, 215–229 (1983).
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Translated from Ukrains'kyi Matematychnyi Zhurnal, Vol. 57, No. 5, pp. 633–643, May, 2005.
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Horbachuk, M.L., Hrushka, Y.I. & Torba, S.M. Direct and Inverse Theorems in the Theory of Approximation by the Ritz Method. Ukr Math J 57, 751–764 (2005). https://doi.org/10.1007/s11253-005-0225-4
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DOI: https://doi.org/10.1007/s11253-005-0225-4