Unbounded Widom factors for orthogonal and residual polynomials

Abstract

We consider Widom factors for two extremal problems on compact sets $K\subset\mathbb R$: the $L^{2}$ Widom factors $W_{2,n}(\mu_{K})$ of monic orthogonal polynomials with respect to the equilibrium measure $\mu_{K}$, and the residual Widom factors $W_{\infty,n}^{(x_{0})}(K)$ for residual polynomials normalized at an exterior point $x_{0}\notin K$. Using weakly equilibrium Cantor sets $K(\gamma)$, we prove that for any sequence $(c_{n})$ with subexponential growth there exists a non-polar $K(\gamma)$ such that $W_{2,n}(\mu_{K(\gamma)})\ge c_{n}$ for all $n$. Moreover, for each $x_{0}\in\mathbb R\setminus K(\gamma)$ we obtain power-type lower bounds of the form $W_{\infty,n}^{(x_{0})}(K(\gamma))\geq c_{n}^{\tau_{x_{0}}}$ with $\tau_{x_{0}}\in(0,1)$, holding for all $n$ in unbounded gaps and along a subsequence in bounded gaps.

Keywords

• Widom factors
• Chebyshev polynomials
• orthogonal polynomials
• residual polynomials

Thanks

The author thanks the American Institute of Mathematics for the hospitality and the stimulating environment of the SQuaRE program.

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