Advertisement

Springer Nature is making SARS-CoV-2 and COVID-19 research free. View research | View latest news | Sign up for updates

Problem of correctness of the best approximation in the space of continuous functions

Abstract

Let WrH w be the subclass of those functions of Cr[a, b], for which ω(f (r),δ)⩽ω(δ), where ω(δ) is a given modulus of continuity, and Pn be the space of algebraic polynomials of degree at most n and πn(f) be the polynomial of best approximation for f(x) on [a, b]. Estimates for

and moduli of continuity of the operators of best approximation on WrH w are established. For example, if ω(δ)=δα, then

This is a preview of subscription content, log in to check access.

Literature cited

  1. 1.

    S. N. Bernshtein, Extremal Properties of Polynomials [in Russian], ONTI, Moscow-Leningrad (1937).

  2. 2.

    G. Freud, “Eine Ungleichung für Tschebysheffsehe Approximationspolynome,” Acta Sci. Math.,19, Nos. 3–4, 162–164 (1958).

  3. 3.

    D. J. Newman and H. S. Shapiro, “Some theorems on Čebyšev Approximation,” Duke Math. J.,30, 673–681 (1963).

  4. 4.

    P. V. Galkin, “On the modulus of continuity of the operator of best approximation in the space of continuous functions,” Mat. Zametki,10, No. 6, 601–613 (1971).

  5. 5.

    V. N. Gabushin, “Inequalities for the norms of a function and its derivatives in the Lp-metrics,” Mat. Zametki,1, No. 3, 291–298 (1967).

  6. 6.

    S. B. Stechkin, “Inequalities between the norms of the derivatives of an arbitrary function,” Acta Sci. Math.,26, Nos. 3–4, 225–230 (1965).

Download references

Author information

Additional information

Translated from Matematicheskie Zametki, Vol. 23, No. 3, pp. 351–360, March, 1978.

The author thanks S. B. Stechkin for the formulation of the problem and assistance with the article.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Kolushov, A.V. Problem of correctness of the best approximation in the space of continuous functions. Mathematical Notes of the Academy of Sciences of the USSR 23, 190–195 (1978). https://doi.org/10.1007/BF01651430

Download citation

Keywords

  • Continuous Function
  • Algebraic Polynomial