Abstract
For Sobolev classes of periodic functions of one variable with restrictions on higher derivatives in L 2, we determine the exact orders of relative widths characterizing the best approximation of a fixed set by its sections of given dimension in the spaces L q.
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REFERENCES
A. N. Kolmogoroff, “Ñber die beste Annäherung von Funktionen einer gegebenen Funktionen Klasse,” Ann. Math., 37, No. 1, 107–110 (1936).
N. P. Korneichuk, Extremal Problems in Approximation Theory [in Russian], Nauka, Moscow (1976).
V. M. Tikhomirov, Some Problems in Approximation Theory [in Russian], Moscow University, Moscow (1976).
A. Pinkus, n-Widths in Approximation Theory, Springer, Berlin (1985).
N. P. Korneichuk, Exact Constants in Approximation Theory [in Russian], Nauka, Moscow (1987).
V. M. Tikhomirov, “Approximation theory,” in: VINITI Series in Contemporary Problems of Mathematics. Fundamental Trends [in Russian], Vol. 14, VINITI, Moscow (1987), pp. 103–260.
V. N. Konovalov, “Estimates of Kolmogorov widths for classes of differentiable periodic functions,” Mat. Zametki, 35, No. 3, 369–380 (1984).
S. B. Stechkin, “On the best approximation of given classes of functions by arbitrary polynomials,” Usp. Mat. Nauk, 9, No. 1, 133–134 (1954).
V. M. Tikhomirov, “Widths of sets in functional spaces and theory of best approximations,” Usp. Mat. Nauk, 15, No. 3, 81–120 (1960).
V. M. Tikhomirov, “Best methods for approximation and interpolation of differentiable functions in the space C [ -1, 1 ],” Mat. Sb., 80, No. 2, 290–304 (1969).
V. M. Tikhomirov, “Some remarks on relative diameters,” Banch Center Publ., 22, 471–474 (1989).
V. F. Babenko, “Approximations in the mean with restrictions imposed on the derivatives of approximating functions,” in: Problems of Analysis and Approximations [in Russian], Institute of Mathematics, Ukrainian Academy of Sciences, Kiev (1989), pp. 9–18.
V. F. Babenko, “On the relative widths of classes of functions with bounded mixed derivative,” East. J. Approxim., 2, No. 3, 319–330 (1996).
V. F. Babenko, “On the best uniform approximations by splines with restrictions imposed on their derivatives,” Mat. Zametki, 50, No. 6, 24–30 (1991).
V. F. Babenko, “On the best L 1-approximations by splines with restrictions imposed on their derivatives,” Mat. Zametki, 51, No. 5, 12–19 (1992).
Yu. N. Subbotin, “Inheritance of the properties of monotonicity and convexity under local approximation,” Zh. Vychisl. Mat. Mat. Fiz., 33, 996–1003 (1993).
Yu. N. Subbotin and S. A. Telyakovskii, “Exact values of relative widths of classes of differentiable functions,” Mat. Zametki, 65, No. 6, 871–879 (1999).
Yu. N. Subbotin and S. A. Telyakovskii, “Relative widths of classes of differentiable functions in the metric of L 2,” Usp. Mat. Nauk, 56, No. 4, 159–160 (2001).
L. É. Azar, Approximation of Functional Classes in the Presence of Restrictions [in Russian], Author's Abstract of the Candidate-Degree Thesis (Physics and Mathematics), Dnepropetrovsk (1999).
R. S. Ismagilov, “On n-dimensional widths of compact sets in a Hilbert space,” Funkts. Anal. Prilozhen., 2, No. 2, 32–39 (1968).
S. M. Nikol'skii, Approximation of Functions of Many Variables and Imbedding Theorems [in Russian], Nauka, Moscow (1969).
V. K. Dzyadyk, Introduction to the Theory of Uniform Polynomial Approximation of Functions [in Russian], Nauka, Moscow (1977).
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Konovalov, V.N. Approximation of Sobolev Classes by Their Sections of Finite Dimension. Ukrainian Mathematical Journal 54, 795–805 (2002). https://doi.org/10.1023/A:1021635530578
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DOI: https://doi.org/10.1023/A:1021635530578