@article{Dzyubenko_2022, title={An interpolatory estimate for copositive polynomial approximations of continuous functions}, volume={74}, url={https://umj.imath.kiev.ua/index.php/umj/article/view/7103}, DOI={10.37863/umzh.v74i4.7103}, abstractNote={<p>UDC 517.9</p> <p>Under the condition that a function $f$, which is continuous on $[-1,1],$ changes its sign at $s$ points $y_i,$ $-1 &lt; y_{s} &lt; y_{s-1} &lt; \dots &lt; y_1 &lt; 1,$ then for each $n \in \mathbb{N}$ greater than some constant $\mathbb{N}$ depending only on $\min_{i=0, \dots ,s}\{y_i -y_{i+1}\},$ $y_{s+1} := -1,$ $y_0 := 1,$ we construct an algebraic polynomial $P_n$ of degree $\le n$ such that $P_n$ has the same sign as $f$ on $[-1,1],$ in particular, $P_n(y_i) = 0,$ $i = 1,\dots ,s,$ and<br>$$<br>|f(x)-P_n(x)|\le c(s)\,\omega_2(f,\sqrt{1-x^2}/n), \quad x\in[-1,1],<br>$$<br>where $c(s)$ is a constant depending only on $s,$ and $\omega_2(f,\cdot)$ is the second order modulus of smoothness of $f$. <br>Note that in this estimate, which is interpolatory at $\pm 1$ and established by DeVore for the unconstrained approximation, it is not possible, even for the unconstrained approximation, to replace $\omega_2$ with $\omega_k,$ $k&gt;2.$</p&gt;}, number={4}, journal={Ukrains’kyi Matematychnyi Zhurnal}, author={Dzyubenko, G. A.}, year={2022}, month={May}, pages={496 - 506} }