Asymptotic Property of Approximation to $x^α$sgn$x$ by Newman Type Operators

Abstract

The approximation of $|x|$ by rational functions is a classical rational problem. This paper deals with the rational approximation of the function $x^α$sgn$x$, which equals $|x|$ if $α = 1$. We construct a Newman type operator $r_n(x)$ and show $$\mathop{\rm min}\limits_{|x|≤1} \{|x^α{\rm sgn}x − r_n(x)| \} ∼ Cn^{−\frac{α}{2}}e^{−\sqrt{2nα}},$$ where $C$ is a constant depending on $α$.