On one property of the modulus of continuity for periodic functions of higher orders

  • Yu. A. Maksymenkova Institute Of Industrial And Business Technologies Ukrainian State University Of Science And Technologies
  • Т. F. Michaylova Institute Of Industrial And Business Technologies Ukrainian State University Of Science And Technologies

Abstract

UDC 517.5

For the moduli of continuity of $2\pi$-periodic functions $\omega_k(f,h)$ of order $k = 1,2,\ldots, $ we prove the inequalities
$$
\omega_k(f,\pi)\leq\frac{2^k}{C_k^{[\frac{k}{2}]}}\frac{1}{\pi}\int\limits_0^{\pi}\omega_k(f,h)dh,
$$
for even $k.$
The inequalities are exact in the spaces $C_{2\pi}$ and $L_1[-\pi,\pi]$.

Author Biography

Т. F. Michaylova , Institute Of Industrial And Business Technologies Ukrainian State University Of Science And Technologies

 

 

References

S. M. Nikol'skij, Ryad Fur'e funkcij s dannym modulem nepreryvnosti, Dokl. AN SSSR, 52, 191 – 193 (1946).

V. A. YUdin, O module nepreryvnosti v $L_2$, Sib. mat. zhurn., 20, 449 – 450 (1979). DOI: https://doi.org/10.1007/BF00970049

S. V. Konyagin, O modulyah nepreryvnosti funkcij, Tez. dokl. Vsesoyuz. shkoly po teorii funkcij (Kemerovo, 1983) (1983), s. 59.

V. I. Ivanov, O module nepreryvnosti v $L_p$, Mat. zametki, 41, № 5, 682 – 686 (1987).

N. M. Ryzhik, I. S. Gradshtejn, Tablicy integralov, summ, ryadov i proizvedenij, Gostekhteorizdat, Moskva (1951).

Published
04.10.2022
How to Cite
Maksymenkova , Y. A., and Michaylova Т. F. “On One Property of the Modulus of Continuity for Periodic Functions of Higher Orders ”. Ukrains’kyi Matematychnyi Zhurnal, Vol. 74, no. 8, Oct. 2022, pp. 1149 -52, doi:10.37863/umzh.v74i8.7217.
Section
Short communications