论文标题
Paley-Wiener空间的点评估
Point evaluation in Paley--Wiener spaces
论文作者
论文摘要
我们研究了Paley的起源 - Wiener Space $ pw^p $ $ 0 <p <\ infty $，i。 e。，我们搜索最小的正常数$ c $，称为$ \ mathscr {c} _p $，使得不等式$ | f（0）|^p \ leq c \ leq c \ | f \ | _p^p $在$ pw^p $中的每个$ f $ cubs中都保留。我们提供证据并证明了几个支持以下单调性猜想的结果：功能$ p \ mapsto \ mathscr {c} _p/p $严格降低了半行$（0，\ infty）$。我们的主要结果意味着$ \ mathscr {c} _p <p/2 $ for $ 2 <p <\ infty $，我们从数字上验证$ \ m mathscr {c} _p> p/2 $ for $ 1 \ leq p <2 $。我们还估计$ \ mathscr {c} _p $的渐近行为为$ p \ to \ infty $，as $ p \ to 0^+$。我们的方法基于表达$ \ Mathscr {C} _p $作为极端问题的解决方案。所有$ 0 <p <\ infty $都存在极端功能；它们是真正的全部功能，只有真实的零功能，并且以$ 1 \ leq p <\ infty $的价格是唯一的唯一功能。在Hörmander和Bernhardsson的工作之后，我们依赖于与极端函数的零相关的某些正交关系，以及某些代表起源的极端功能和一般函数的某些积分公式。我们还为Landau- Pollak-Slepian运算符的最大特征值（频率浓度）使用精确的数值估计值。建立了许多关于极端函数零零分布的定性和定量结果。在$ 1 <p <\ infty $的范围内，与极值功能的零相关的正交关系与de branges空间有关。我们陈述了许多猜想和与$ \ mathscr {c} _p $和极端功能有关的进一步的开放问题。
We study the norm of point evaluation at the origin in the Paley--Wiener space $PW^p$ for $0 < p < \infty$, i. e., we search for the smallest positive constant $C$, called $\mathscr{C}_p$, such that the inequality $|f(0)|^p \leq C \|f\|_p^p$ holds for every $f$ in $PW^p$. We present evidence and prove several results supporting the following monotonicity conjecture: The function $p\mapsto \mathscr{C}_p/p$ is strictly decreasing on the half-line $(0,\infty)$. Our main result implies that $\mathscr{C}_p <p/2$ for $2<p<\infty$, and we verify numerically that $\mathscr{C}_p > p/2$ for $1 \leq p < 2$. We also estimate the asymptotic behavior of $\mathscr{C}_p$ as $p \to \infty$ and as $p \to 0^+$. Our approach is based on expressing $\mathscr{C}_p$ as the solution of an extremal problem. Extremal functions exist for all $0<p<\infty$; they are real entire functions with only real zeros, and the extremal functions are known to be unique for $1\leq p < \infty$. Following work of Hörmander and Bernhardsson, we rely on certain orthogonality relations associated with the zeros of extremal functions, along with certain integral formulas representing respectively extremal functions and general functions at the origin. We also use precise numerical estimates for the largest eigenvalue of the Landau--Pollak--Slepian operator of time--frequency concentration. A number of qualitative and quantitative results on the distribution of the zeros of extremal functions are established. In the range $1<p<\infty$, the orthogonality relations associated with the zeros of the extremal function are linked to a de Branges space. We state a number of conjectures and further open problems pertaining to $\mathscr{C}_p$ and the extremal functions.
