Abstract
The exact exponent of complexity is found for approximate solutions of a certain class of operator equations in a Hilbert space. A method for information setup and the algorithm for realization of this optimal degree are presented. As a consequence, we find the exact exponent of complexity for approximate solutions of Fredholm integral equations of the second kind whose kernels and free terms include square integrable ψ-derivatives.
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J. F. Traub and H. Wozniakowski,A General Theory of Optimal Algorithms [Russian translation], Mir, Moscow (1983).
J. F. Traub, G. W. Wasilkowskii, and H. Wozniakowski,Information, Uncertainty, Complexity [Russian translation], Mir, Moscow (1988).
L. V. Kantorovich and G. P. Akilov,Functional Analysis [in Russian], Nauka, Moscow (1977).
S. V. Pereverzev and K. Sh. Makhkamov, “Galerkin information, hyperbolic cross, and the complexity of operator equations,”Ukr. Mat. Zh.,43, No. 5, 639–648 (1991).
K. I. Babenko,Foundations of Numerical Analysis [in Russian], Nauka, Moscow (1986).
V. M. Tikhomirov,Problems of the Theory of Approximation [in Russian], Moscow University, Moscow (1976).
A. I. Stepanets,Classification and Approximation of Periodic Functions, Kluwer, Dordrecht (1995).
S. V. Pereverzev, “Hyperbolic cross and the complexity of approximate solution of Fredholm integral equations of the second kind with differentiable kernels,”Sib. Mat. Zh.,32, No. 1, 107–115 (1991).
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Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 46, No. 7, pp. 893–903, July, 1994.
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Makhkamov, K.S. On the exact degree of complexity of a class of operator equations of the second kind in a Hilbert space. Ukr Math J 46, 979–990 (1994). https://doi.org/10.1007/BF01056675
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DOI: https://doi.org/10.1007/BF01056675