Entire Approximations to the Truncated Powers

Abstract

For a variation diminishing function $g$ which is analytic on a set containing the real line and any real polynomial $P$, we prove that $g+P$ has at most $\text{deg}(P)+2$ real zeros. Based on this estimate, we present a way to construct entire approximations $G_n$ to the truncated powers $x_+^n$ for $n\in{\bf N}_0$. Here $x_+^n=x^n$ for $x>0$ and $x_+^n=0$ for $x<0$. The function $G_n$ is constructed in such a way that \[ G_n(x)-x_+^n=F(x)H_n(x)\] holds, where $F$ is entire and $H_n$ has no zeros on the real line. The function $G_n$ can be viewed as an interpolant of $x_+^n$ with a nodal set that is given by the (real) zeros of $F$. As an application of this method, we give explicit formulas for best $L^1({\bf R})$-approximation and best one-sided $L^1({\bf R})$-approximation from the class of entire functions with given exponential type $\eta$ to $x_+^n$. These approximations are given in terms of the logarithmic derivative of the Euler Gamma function.