帕利-维纳空间中的最优集中
Optimal concentration in the Paley-Wiener space
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中文总结 AI 辅助
研究帕利-维纳空间中函数在有限测度集\(E\)上的\(L^{2}\)质量集中问题，基于圆上解析三角多项式再生核的普适性极限，分两步证明区间能优化集中，得出区间在该空间中优化集中的结论。
中文摘要 AI 辅助
设\(\Omega \subset \mathbb{R}\)为有界区间，\(PW(\Omega )\)为相应的帕利-维纳空间。对于有限测度的可测集\(E\subset \mathbb{R}\)，考虑\(PW(\Omega )\)中函数在\(E\)中的\(L^{2}\)质量的最大可能比例。我们证明这种集中不大于在测度为\(\lvert E\rvert\)的区间上达到的集中。即区间在带限函数的帕利-维纳空间中优化集中。证明基于圆上解析三角多项式再生核的一种普适性类型极限，分两步。首先，建立圆上解析三角多项式的最优集中定理。其次，通过控制其投影核为帕利-维纳辛格核的中点黎曼和的圆的展开，普适性类型极限将结果从圆转移到实直线。
英文摘要
Let $Ω\subset \mathbb{R}$ be a bounded interval and let $PW(Ω)$ be the corresponding Paley--Wiener space. For a measurable set $E\subset \mathbb{R}$ of finite measure, consider the largest possible fraction of the $L^{2}$-mass of a function in $PW(Ω)$ that can lie in $E$. We prove that this concentration is no larger than the concentration attained on an interval of measure $\lvert E\rvert $. Thus, \emph{intervals optimize concentration in the Paley-Wiener space of band-limited functions.} The proof, based on an universality-type limit of the reproducing kernel of analytic trigonometric polynomials on the circle, has two steps. First, we establish an \emph{optimal concentration theorem for analytic trigonometric polynomials on the circle}. Second, the universality-type limit transfers the result from the circle to the real line, by controlling the expansion of circles whose projection kernels are midpoint Riemann sums for the Paley--Wiener sinc kernel.
