A four-way Szeg\H{o} theorem for $L^p$ extremal polynomials on subsets of $\mathbb R$

Abstract

We prove a four-way Szeg\H{o} theorem for $L^p$ extremal polynomials on compact supports $K=K_0\cup X\subset\mathbb R$, where $K_0$ is a regular compact set and $X$ is a finite or countable set of isolated points. For every $2\le p\le\infty$, including the weighted Chebyshev case $p=\infty$ under the corresponding assumptions on the weight, any three of the Parreau--Widom condition for $K_0$, the Blaschke condition for $X$, the Szeg\H{o} condition for the weight, and the Widom condition $0<\limsup_{n\rightarrow\infty}W_{p,n}<\infty$ imply the fourth. As a consequence, for every regular compact set $K\subset\mathbb R$ of positive capacity, the Parreau--Widom condition is equivalent both to boundedness of the unweighted Chebyshev Widom factors and to boundedness of the equilibrium-measure $L^2$ Widom factors. For every $0<p\le\infty$, we also prove upper and lower bounds for the Widom factors in which the contributions of the weight, the isolated points, and the gaps of $K_0$ appear separately. Finally, we give examples illustrating the sharpness of our results. For $p=2$, we realize every combination of the following five properties that is not excluded by the implications proved in this work: the Parreau--Widom condition, the Blaschke condition, the Szeg\H{o} condition, boundedness of the Widom factors from above, and boundedness of the Widom factors away from zero.