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arXiv 2107.13245math.CA
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广义Jacobi测度的广义因子

Widom factors for generalized Jacobi measures

Gökalp Alpan

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摘要：

我们研究了与Chebyshev多项式相关的Widom因子$W_{\infty,n}(K,w)$在$[-1,1]$的紧子集上的权值$w(x)=\sqrt{1+x}$和$w(x)=\sqrt{1-x}$的最优上下界。我们显示哪些集饱和这些界限。我们考虑$L_2(μ)$极值多项式的Widom因子$dμ(x)=(1-x)^α(1+x)^βdμ_K(x)$，其中$α+β\geq 1$，$α,β\in\mathbb{N}\cup \{0\}$和$μ_K$是$[-1,1]$中的紧正则集$K$与$\pm 1\in K$的平衡测度。我们表明，对于这样的措施，改进的下限$[W_{2,n}(μ)]^2\geq 2S(μ)$举行。对于特殊情况$dμ(x)=(1-x^2)dμ_K(x)$，$dμ(x)=(1-x)dμ_K(x)$，$dμ(x)=(1+x)dμ_K(x)$，我们确定哪些集饱和这个下限，并讨论如何饱和下限$[W_{2,n}(μ)]^2$和$W_{\infty,n}(K,w)$相关。

英文摘要：

We study optimal lower and upper bounds for Widom factors $W_{\infty,n}(K,w)$ associated with Chebyshev polynomials for the weights $w(x)=\sqrt{1+x}$ and $w(x)=\sqrt{1-x}$ on compact subsets of $[-1,1]$. We show which sets saturate these bounds. We consider Widom factors $W_{2,n}(μ)$ for $L_2(μ)$ extremal polynomials for measures of the form $dμ(x)=(1-x)^α(1+x)^βdμ_K(x)$ where $α+β\geq 1$, $α,β\in\mathbb{N}\cup \{0\}$ and $μ_K$ is the equilibrium measure of a compact regular set $K$ in $[-1,1]$ with $\pm 1\in K$. We show that for such measures the improved lower bound $[W_{2,n}(μ)]^2\geq 2S(μ)$ holds. For the special cases $dμ(x)=(1-x^2)dμ_K(x)$, $dμ(x)=(1-x)dμ_K(x)$, $dμ(x)=(1+x)dμ_K(x)$ we determine which sets saturate this lower bound and discuss how saturated lower bounds for $[W_{2,n}(μ)]^2$ and $W_{\infty,n}(K,w)$ are related.

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