On the nonsymmetric approximation of continuous functions in the integral metric

V.F. Babenko; O.V. Polyakov

doi:10.15421/241909

On the nonsymmetric approximation of continuous functions in the integral metric

V.F. Babenko (Oles Honchar Dnipro National University), https://orcid.org/0000-0001-6677-1914
O.V. Polyakov (Public Education Institution "Dnipropetrovsk Regional Physical and Mathematical Boarding Lyceum")

Abstract

In the paper, an exact estimate of the best nonsymmetric approximation in the integral metric by the constants of continuous functions that belong to the classes $$$H^\omega[a,b]$$$ is proved.
Taking into account Babenko's theorem on the connection of nonsymmetric approximation with the usual best approximation in the integral metric and the best one-sided approximations, from the proved result we obtain the exact estimate for the usual best approximation obtained by N.P. Korneichuk, and the exact estimate for the best one-sided approximation obtained by V.G. Doronin and A.A. Ligun.

Keywords

modulus of continuity; nonsymmetric approximations

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References

Babenko V.F. "Non-symmetric approximations in spaces of integrable functions", Ukrainian Math. J., 1982; 34(4); pp. 409-416. (in Russian)

Korneichuk N.P. "On widths of classes of continuous functions in $$$L_p$$$ space", Matem. zametki, 1971; 10(5); pp. 493-500. (in Russian) doi:10.1007/BF01109031

Korneichuk N.P. Splines in approximation theory, Nauka, Moscow, 1984; 544 p. (in Russian)

Khoroshko N.P. "On the best approximation of functions of $$$H_{\omega}[0;1]$$$ class by polynomials by Haar system in $$$L_p$$$ metric", Researches on modern problems of summation and approximation of functions and their applications, DGU, Dnipropetrovsk, 1972; pp. 74-76. (in Russian)

Doronin V.G., Ligun A.A. "On the problem of the best approximation of some classes of continuous functions", Researches on modern problems of summation and approximation of functions and their applications, DGU, Dnipropetrovsk, 1974; pp. 42-49. (in Russian)

Korneichuk N.P. Extremum problems in approximation theory, Moscow, 1976; 320 p. (in Russian)