论文标题
在$ l_p $ tormutal多项式的宽度因素上
On the Widom factors for $L_p$ extremal polynomials
论文作者
论文摘要
我们继续研究[4]中启动的$ L_P（μ）$极端多项式的Widom因子。在这项工作中，我们表征了[4]中获得的下限饱和的集合，建立了相对于度量$μ$的宽度因子的连续性，并表明，尽管下限$ [w_ {2，n}（μ_K）（μ_k）]^2 \ geq 2s（μ_k）的平衡级$ $ undody undound undound undound undound compact $ k $ k $ k $ k $ k $ k $ k k k k k k in compact $ k k k in $ [w_ {p，n}（μ）]^p \ geq s（μ）$即使对于度量$dμ=wdμ_k$，带有多项式重量$ w $ $ k \ subset \ subset \ mathbb r $。我们还研究了[16，23]中引入的多项式预示图中的撤回措施，并获得了此类措施的宽度因子的不变性。最后，我们详细研究了正交多项式相对于圆弧上的平衡度量的宽度因子，尤其是找到它们的极限，最大最小和至上，并表明它们严格地随着程度而严格地增加了单调，并且严格随着ARC的长度而严格降低单位酮。
We continue our study of the Widom factors for $L_p(μ)$ extremal polynomials initiated in [4]. In this work we characterize sets for which the lower bounds obtained in [4] are saturated, establish continuity of the Widom factors with respect to the measure $μ$, and show that despite the lower bound $[W_{2,n}(μ_K)]^2\geq 2S(μ_K)$ for the equilibrium measure $μ_K$ on a compact set $K\subset\mathbb R$ the general lower bound $[W_{p,n}(μ)]^p\geq S(μ)$ is optimal even for measures $dμ=wdμ_K$ with polynomial weights $w$ on $K\subset\mathbb R$. We also study pull-back measures under polynomial pre-images introduced in [16, 23] and obtain invariance of the Widom factors for such measures. Lastly, we study in detail the Widom factors for orthogonal polynomials with respect to the equilibrium measure on a circular arc and, in particular, find their limit, infimum, and supremum and show that they are strictly monotone increasing with the degree and strictly monotone decreasing with the length of the arc.
