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arXiv 0910.4223math.CAmath-phmath.MP
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平面正交多项式的范数和根及变权的$L^p$ -最优多项式

On the norms and roots of orthogonal polynomials in the plane and $L^p$-optimal polynomials with respect to varying weights

F. Balogh, M. Bertola

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摘要：

对于复平面上的一个子集的措施，我们认为$L^p$ -最佳加权多项式，即一元多项式的程度$n$与不同的重量的形式$w^n = {\rm e}^{-n V}$，最小化的$L^p$ -规范，$1 \leq p \leq \infty$。结果表明，最终所有，但一致有界的数量的根的$L^p$ -最佳多项式位于一个小的邻域内的支持一定的平衡措施;渐近的$n$ TH根的$L^p$规范也提供。$p=\infty$的情况是众所周知的，对应于加权切比雪夫多项式; $p=2$的情况对应于正交多项式以及任何其他$1\leq p <\infty$是我们的贡献。

英文摘要：

For a measure on a subset of the complex plane we consider $L^p$-optimal weighted polynomials, namely, monic polynomials of degree $n$ with a varying weight of the form $w^n = {\rm e}^{-n V}$ which minimize the $L^p$-norms, $1 \leq p \leq \infty$. It is shown that eventually all but a uniformly bounded number of the roots of the $L^p$-optimal polynomials lie within a small neighborhood of the support of a certain equilibrium measure; asymptotics for the $n$th roots of the $L^p$ norms are also provided. The case $p=\infty$ is well known and corresponds to weighted Chebyshev polynomials; the case $p=2$ corresponding to orthogonal polynomials as well as any other $1\leq p <\infty$ is our contribution.

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