Fourier Uncertainty Principles, Scale Space Theory and the Smoothest Average

Abstract

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 $\alpha > 0$ and $\beta > n/2$, there exists $c_{\alpha, \beta,n} > 0$ such that for all $u \in L^1(\mathbb{R}^n)$ $$ \| |\xi|^{\beta} \cdot \widehat{u}\|^{\alpha}_{L^{\infty}(\mathbb{R}^n)} \cdot \| |x|^{\alpha} \cdot u \|^{\beta}_{L^1(\mathbb{R}^n)} \geq c_{\alpha, \beta,n} \|u\|_{L^1(\mathbb{R}^n)}^{\alpha + \beta}.$$ For $\beta = 1$, any nonnegative extremizer of the inequality serves as the best averaging function in the sense above, $\beta \neq 1$ corresponds to other derivatives. For $(n, \beta)=(1,1)$ we use the Shannon-Whittaker formula to prove that the characteristic function $u(x) = \chi_{[-1/2,1/2]}$ is a local minimizer among functions defined on $[-1/2,1/2]$ for $\alpha \in \left\{2,3,4,5,6\right\}$. We provide a sufficient condition for general $\alpha$ in terms of a sign pattern for the hypergeometric function $_1F_2$.