Abstract
A definition of the Hausdorff alternance is given. In its terms we give a sufficient condition for an algebraic polynomial to have minimal deviation from a function f in the Hausdorff α-metric. A condition under which a polynomial Pn is the unique best-approximation polynomial for a function f and a necessary condition for Pn to have minimal deviation from f are given. Similar theorems for 2π-periodic functions are formulated. Bibliography: 3 titles.
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Literature Cited
B. Sendov,Hausdorff Approximations [in Russian], Sofia (1979).
B. D. Boyanov, “On best-approximation polynomials with respect to the Hausdorff distance,”Godishn. Sof. Univ., Mat. Fak. 64, 161–170 (1971).
A. S. Andreev, “Stability of best approximations with respect to the Hausdorff distance,” in:Theory of Approximation of Functions [in Russian], Moscow (1977), pp. 12–13.
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Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 217, 1994, pp. 130–143.
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Petukhov, A.P. On best-approximation polynomials in the Hausdorff metric. J Math Sci 85, 1839–1848 (1997). https://doi.org/10.1007/BF02355294
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DOI: https://doi.org/10.1007/BF02355294