An interpolatory estimate for copositive polynomial approximations of continuous functions

Abstract

UDC 517.9

Under the condition that a function $f$, which is continuous on $[-1,1],$ changes its sign at $s$ points $y_i,$ $-1 < y_{s} < y_{s-1} < \dots < y_1 < 1,$ 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
$$
|f(x)-P_n(x)|\le c(s)\,\omega_2(f,\sqrt{1-x^2}/n), \quad x\in[-1,1],
$$
where $c(s)$ is a constant depending only on $s,$ and $\omega_2(f,\cdot)$ is the second order modulus of smoothness of $f$.
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 $\omega_k,$ $k>2.$