Zero distribution of Müntz extremal polynomials in $L_p[0,1]$

Abstract:

Let $\left \{ \lambda _{j}\right \} _{j=0}^{\infty }$ be a sequence of distinct positive numbers. Let $1\leq p\leq \infty$ and let $T_{n,p}=T_{n,p}\left \{ \lambda _{0},\lambda _{1},\lambda _{2},\dots ,\lambda _{n}\right \} \left ( x\right )$ denote the $L_{p}$ extremal Müntz polynomial in $\left [ 0,1\right ]$ with exponents $\lambda _{0},\lambda _{1},\lambda _{2},\dots ,\lambda _{n}$. We investigate the zero distribution of $\left \{ T_{n,p}\right \} _{n=1}^{\infty }$. In particular, we show that if \begin{equation*} \lim _{n\rightarrow \infty }\frac {\lambda _{n}}{n}=\alpha >0, \end{equation*} then the normalized zero counting measure of $T_{n,p}$ converges weakly as $n\rightarrow \infty$ to \begin{equation*} \frac {\alpha }{\pi }\frac {t^{\alpha -1}}{\sqrt {t^{\alpha }\left ( 1-t^{\alpha } \right ) }}dt, \end{equation*} while if $\alpha =0$ or $\infty$, the limiting measure is a Dirac delta at $0$ or $1$, respectively.