Constructive Approximation

, Volume 22, Issue 2, pp 273–295 | Cite as

Entire Approximations to the Truncated Powers



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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Copyright information

© Springer 2005

Authors and Affiliations

  1. 1.Department of Mathematics, University of British Columbia, Vancouver, BC V6T 1Z2Canada

Personalised recommendations