To the question of approximation of continuous periodic functions by trigonometric polynomials

V.V. Shalaev (Dnipropetrovsk State University)

Abstract


In the paper, it is proved that
$$1 - \frac{1}{2n} \leqslant \sup\limits_{\substack{f \in C\\f \ne const}} \frac{E_n(f)_C}{\omega_2(f; \pi/n)_C} \leqslant \inf\limits_{L_n \in Z_n(C)} \sup\limits_{\substack{f \in C\\f \ne const}} \frac{\| f - L_n(f) \|_C}{\omega_2 (f; \pi/n)_C} \leqslant 1$$
where $$$\omega_2(f; t)_C$$$ is the modulus of smoothness of the function $$$f \in C$$$, $$$E_n(f)_C$$$ is the best approximation by trigonometric polynomials of the degree not greater than $$$n-1$$$ in uniform metric, $$$Z_n(C)$$$ is the set of linear bounded operators that map $$$C$$$ to the subspace of trigonometric polynomials of degree not greater than $$$n-1$$$.

References


Korneichuk N.P. "Exact constant in Jackson theorem on the best uniform appproximation of continuous periodic functions", Dokl. AN SSSR, 1962; 145(3): pp. 514-515. (in Russian)

Chernykh N.I. "On Jackson inequality in $$$L_2$$$ space", Trudy MIAN SSSR, 1967; 88: pp. 71-74. (in Russian)

Davidchik A.N., Ligun A.A. "To Jackson's theorem", Matem. zametki, 1974; 16(6): pp. 681-690. (in Russian) doi:10.1007/BF01149787

Babenko V.F., Pichugov S.A. "On one extremal property of Korovkin's means", Res. Math., 1977; pp. 7-8. (in Russian)

Stechkin S.B. "On approximation of periodic functions by Favard sums", Trudy MIAN SSSR, 1971; 109. (in Russian)

Akhiezer N.I. Lectures on Approximation Theory, 1965. (in Russian)

Timan A.F. Approximation theory for real-variable functions, Fizmatgiz, 1960. (in Russian)




DOI: https://doi.org/10.15421/247711

  

Refbacks

  • There are currently no refbacks.


Copyright (c) 1977 V.V. Shalaev

Creative Commons License
This work is licensed under a Creative Commons Attribution 4.0 International License.


Registered in


ISSN (Online): 2664-5009
ISSN (Print): 2664-4991
DNU