Zero distribution of Müntz extremal polynomials in $L_{p}\left[ 0,1\right]$

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

$\lim_{n\rightarrow \infty }\frac{\lambda _{n}}{n}=\alpha >0,$

then the normalized zero counting measure of $T_{n,p}$ converges weakly as $n\rightarrow \infty$ to

$\frac{\alpha }{\pi }\frac{t^{\alpha -1}}{\sqrt{t^{\alpha }\left( 1-t^{\alpha} \right) }}dt,$

while if $\alpha =0$ or $\infty$, the limiting measure is a Dirac delta at $0$ or $1$, respectively.