Comonotone Second Jackson's Inequality
Let $2s$ points $y_i=-π\le y_{2s}<\ldots 2$, then $$E_n^{(1)}(f;Y)\le \frac c{n^r}, $$ where $c=c(r,Y)=const$ depending only on $r$ and $Y$, ${\Bbb W}^r$ Sobolev space.
math.CA↗
arXiv subjects
M. G. Pleshakov
Publications and source records attributed to M. G. Pleshakov.
One Counterexample of Comonotone Approximation of $2π$-periodic Function on Trigonometric Polynomials
Let $2s$ points $y_i=-π\le y_{2s}<\ldots 3$, and $n\in\Bbb N$ there a function $f(x):=f(x;s,Y,n,k)$ exists, such that $f\in\bigtriangleup^{(1)}(Y)\bigcap{\Bbb C}^{(1)}$ and $$ E_n^{(1)}(f;Y)>B_Yn^{\frac k3 -1}\frac 1nω_k\left(f';\frac 1n\right), $$ where $B_Y=$const, depending only on $Y$ and $k$; $ω_k$ is the modulus of smoothness of order $k$, of $f$.
math.CA↗