论文标题
傅立叶不确定性原理，规模空间理论和最平均平均水平
Fourier Uncertainty Principles, Scale Space Theory and the Smoothest Average
论文作者
论文摘要
令$ f \ in l^{2}（\ mathbb {r}^n）$，假设我们有兴趣以固定比例计算其平均值。这很容易：我们选择概率分布的密度$ u _ {} $，平均0，并在所需的比例下进行一些瞬间，并计算卷积$ u_ {} * f $。 $ u $有一个特别自然的选择吗？这个问题是在规模空间理论中研究的，高斯是一个流行的答案。我们很感兴趣的是，$ u $的规范选择是否可以来自新的公理：固定刻度后，平均应该尽可能少振荡，即$$ u_ {} = \ arg \ arg \ min_ {u_ {} u_ {}}} \ sup_ {f \ \ nabla（u_ {} *f）\ | _ {l^2（\ Mathbb {r}^n）}}}}} {\ | f \ | _ {l^2（\ m mathbb {r}^n）}。存在$ c_ {α，β，n}> 0 $，以使得所有$ u \ in l^1（\ Mathbb {r}^n）$ $ $ $ $ \ | |ξ|^β\ cdot \ wideHat {u} \ |^α_{l^{\ infty}（\ Mathbb {r}^n）} \ cdot \ | |x|^α \cdot u \|^β_{L^1(\mathbb{R}^n)} \geq c_{α, β,n} \|u\|_{L^1(\mathbb{R}^n)}^{α+ β}.$$ For $β= 1$, any nonnegative extremizer of the在上述意义上，不平等是最佳的平均函数，$β\ neq 1 $对应于其他衍生物。 For $(n, β)=(1,1)$ we use the Shannon-Whittaker formula to prove that the characteristic function $u(x) = χ_{[-1/2,1/2]}$ is a local minimizer among functions defined on $[-1/2,1/2]$ for $α\in \left\{2,3,4,5,6\right\}$.在超几何函数$ _1F_2 $的标志模式方面，我们为一般$α$提供了足够的条件。
Let $f \in L^{2}(\mathbb{R}^n)$ and suppose we are interested in computing its average at a fixed scale. This is easy: we pick the density $u_{}$ of a probability distribution with mean 0 and some moment at the desired scale and compute the convolution $u_{} * f$. Is there a particularly natural choice for $u$? This question is studied in scale space theory and the Gaussian is a popular answer. We were interested whether a canonical choice for $u$ can arise from a new axiom: having fixed a scale, the average should oscillate as little as possible, i.e. $$ u_{} = \arg\min_{u_{}} \sup_{f \in L^2(\mathbb{R}^n)} \frac{\| \nabla (u_{} *f) \|_{L^2(\mathbb{R}^n)}}{\|f\|_{L^2(\mathbb{R}^n)}}.$$ This optimal function turns out to be a minimizer of an uncertainty principle: for $α> 0$ and $β> n/2$, there exists $c_{α, β,n} > 0$ such that for all $u \in L^1(\mathbb{R}^n)$ $$ \| |ξ|^β \cdot \widehat{u}\|^α_{L^{\infty}(\mathbb{R}^n)} \cdot \| |x|^α \cdot u \|^β_{L^1(\mathbb{R}^n)} \geq c_{α, β,n} \|u\|_{L^1(\mathbb{R}^n)}^{α+ β}.$$ For $β= 1$, any nonnegative extremizer of the inequality serves as the best averaging function in the sense above, $β\neq 1$ corresponds to other derivatives. For $(n, β)=(1,1)$ we use the Shannon-Whittaker formula to prove that the characteristic function $u(x) = χ_{[-1/2,1/2]}$ is a local minimizer among functions defined on $[-1/2,1/2]$ for $α\in \left\{2,3,4,5,6\right\}$. We provide a sufficient condition for general $α$ in terms of a sign pattern for the hypergeometric function $_1F_2$.
