Geodesic
Sharp Constant in Jackson's Inequality with Modulus of Smoothness for Uniform Approximations of Periodic Functions
Matematičeskie zametki, Tome 93 (2013) no. 6, pp. 932-938

Voir la notice de l'article provenant de la source Math-Net.Ru

It is proved that, in the space $\mathrm{C}_{2\pi}$, for all $k,n\in\mathbb N$, $n>1$, the following inequalities hold: $$ \biggl(1-\frac {1}{2n}\biggr)\frac{k^2+1}{2}\le \sup_{\substack{f\in \mathrm{C}_{2\pi}\\ f\ne\mathrm{const}}} \frac{{e}_{n-1}(f)}{\omega_2(f,\pi/(2nk))}\le \frac{k^2+1}{2}\mspace{2mu}. $$ where ${e}_{n-1}(f)$ is the value of the best approximation of $f$ by trigonometric polynomials and $\omega_2(f,h)$ is the modulus of smoothness of $f$. A similar result is also obtained for approximation by continuous polygonal lines with equidistant nodes.
Keywords: Jackson's inequality, periodic function, trigonometric polynomial, modulus of smoothness, polygonal line, Steklov mean
Mots-clés : Favard sum. 
@article{MZM_2013_93_6_a10,
     author = {S. A. Pichugov},
     title = {Sharp {Constant} in {Jackson's} {Inequality} with {Modulus} of {Smoothness} for {Uniform} {Approximations} of {Periodic} {Functions}},
     journal = {Matemati\v{c}eskie zametki},
     pages = {932--938},
     publisher = {mathdoc},
     volume = {93},
     number = {6},
     year = {2013},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/MZM_2013_93_6_a10/}
}
TY  - JOUR
AU  - S. A. Pichugov
TI  - Sharp Constant in Jackson's Inequality with Modulus of Smoothness for Uniform Approximations of Periodic Functions
JO  - Matematičeskie zametki
PY  - 2013
SP  - 932
EP  - 938
VL  - 93
IS  - 6
PB  - mathdoc
UR  - http://geodesic.mathdoc.fr/item/MZM_2013_93_6_a10/
LA  - ru
ID  - MZM_2013_93_6_a10
ER  - 
%0 Journal Article
%A S. A. Pichugov
%T Sharp Constant in Jackson's Inequality with Modulus of Smoothness for Uniform Approximations of Periodic Functions
%J Matematičeskie zametki
%D 2013
%P 932-938
%V 93
%N 6
%I mathdoc
%U http://geodesic.mathdoc.fr/item/MZM_2013_93_6_a10/
%G ru
%F MZM_2013_93_6_a10
S. A. Pichugov. Sharp Constant in Jackson's Inequality with Modulus of Smoothness for Uniform Approximations of Periodic Functions. Matematičeskie zametki, Tome 93 (2013) no. 6, pp. 932-938. http://geodesic.mathdoc.fr/item/MZM_2013_93_6_a10/