论文标题
关于$ l^p $球的傅立叶变换的评论
A remark on the Fourier transform of $l^p$ balls
论文作者
论文摘要
我们通过一个示例重新检查了集合边界的曲率与其特征函数的傅立叶变换的衰减之间的连接。令$ b_p \ subset \ mathbb {r}^2 $表示$ \ mathbb {r}^2 $ in $ l^p $ -norm中的单位球。这是Hlawka经典结果的结果，即（1,2] $中的每个$ p \，存在$ c（p）> 0 $，因此$ | \widehatχ_{b_p}（b_p}（ω） \ frac {c（p）} {|ω|^{3/2}} \ quad（ω\ in \ mathbb {r}^2，| |ω| \ text {light}） $ c（p）$的渐近行为为$ p \ rightarrow1+$。
We re-examine through an example the connection between the curvature of the boundary of a set, and the decay at infinity of the Fourier transform of its characteristic function. Let $B_p\subset\mathbb{R}^2$ denote the unit ball of $\mathbb{R}^2$ in the $l^p$-norm. It is a consequence of a classical result of Hlawka that for each $p\in(1,2]$, there exists $C(p)>0$ such that $$ |\widehatχ_{B_p}(ω)|\le \frac{C(p)}{|ω|^{3/2}}\quad(ω\in\mathbb{R}^2, |ω|\text{ large}). $$ The above estimate does not hold for $p=1$. Thus, one expects that $C(p)\rightarrow\infty$ as $p\rightarrow1+$; we determine the sharp asymptotic behaviour of $C(p)$ as $p\rightarrow1+$.
