Abstract
We obtain estimates for the approximation of functions of the class W r H ω, where ω(t) is a convex modulus of continuity such that tω′(t) does not decrease, by algebraic polynomials with regard for the position of a point on the segment [−1, 1]. The estimates obtained cannot be improved for all moduli of continuity simultaneously.
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REFERENCES
S. M. Nikol'skii, “On the best polynomial approximation of functions satisfying the Lipschitz condition,” Izv. Akad. Nauk SSSR, Ser. Mat. 10, 295–322 (1946).
N. P. Korneichuk and A. I. Polovina, “On the approximation of continuous and differentiable functions by algebraic polynomials on a segment,” Dokl. Akad. Nauk SSSR 166, No. 2, 281–283 (1966).
N. P. Korneichuk and A. I. Polovina, “On the approximation of functions satisfying the Lipschitz condition by algebraic polynomials,” Mat. Zametki 9, No. 4, 441–447 (1971).
N. P. Korneichuk and A. I. Polovina, “On the approximation of continuous functions by algebraic polynomials,” Ukr. Mat. Zh. 24, No. 3, 328–340 (1972).
A. A. Ligun, “On the best approximation of differentiable functions by algebraic polynomials,” Izv. Vyssh. Uchebn. Zaved., Ser. Mat. No. 4, 53–60 (1980).
V. N. Temlyakov, “Approximation of functions of the class \(W_\infty ^1 \) by algebraic polynomials,” Mat. Zametki 29, No. 4, 597–602 (1981).
Yu. A. Brudnyi, “On A. F. Timan's papers on the polynomial approximation of functions,” in: Proceedings of the All-Union Conference on the Theory of Approximation of Functions [in Russian], Dnepropetrovsk (1991), pp. 13–17.
R. M. Trigub, “Direct theorems on approximation of functions smooth on a segment by algebraic polynomials,” Mat. Zametki 54, No. 6, 113–121 (1993).
R. M. Trigub, “Approximation of functions by polynomials with integer coefficients,” Izv. Akad. Nauk SSSR, Ser. Mat. 26, No. 2, 261–280 (1962).
V. P. Motornyi, “Approximation of fractional-order integrals by algebraic polynomials,” Ukr. Mat. Zh. 51, No. 7, 940–951 (1999).
N. P. Korneichuk, “On the best uniform approximation on certain classes of continuous functions,” Dokl. Akad. Nauk SSSR 140, 748–751 (1961).
N. P. Korneichuk, “On the best approximation of continuous functions,” Izv. Akad. Nauk SSSR, Ser. Mat. 27, 29–44 (1963).
A. V. Pokrovskii, “On one theorem of A. F. Timan,” Funkts. Anal. Primen. 1, No. 3, 93–94 (1967).
S. B. Stechkin, “On the order of the best approximations of continuous functions,” Izv. Akad. Nauk SSSR, Ser. Mat. 15, 219–242 (1951).
A. F. Timan, Theory of Approximation of Functions of a Real Variable [in Russian], Fizmatgiz, Moscow (1960).
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Motornyi, V.P. On Exact Estimates for the Pointwise Approximation of the Classes WrHω by Algebraic Polynomials. Ukrainian Mathematical Journal 53, 916–937 (2001). https://doi.org/10.1023/A:1013399801684
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DOI: https://doi.org/10.1023/A:1013399801684