Fourier uncertainty principles, scale space theory and the smoothest average

Let $f \in L^{2}(\mathbb{R}^n)$ and suppose we are interested in computing its average at a fixed scale. This is easy: pick the density $u_{}$ of probability distribution with mean 0 some moment desired scale compute convolution $u_{} * f$. Is there particularly natural choice for $u$? question studied space theory Gaussian popular answer. We were whether canonical $u$ can arise from new axiom: having scale, should oscillate as little possible, i.e. $$ u_{} = \arg\min_{u_{}} \sup_{f L^2(\mathbb{R}^n)} \frac{\| \nabla (u_{} *f) \|_{L^2(\mathbb{R}^n)}}{\|f\|_{L^2(\mathbb{R}^n)}}.$$ optimal function turns out to be minimizer an uncertainty principle: $\alpha > 0$ $\beta n/2$, exists $c_{\alpha, \beta,n} such that all $u L^1(\mathbb{R}^n)$ \| |\xi|^{\beta} \cdot \widehat{u}\|^{\alpha}_{L^{\infty}(\mathbb{R}^n)} |x|^{\alpha} u \|^{\beta}_{L^1(\mathbb{R}^n)} \geq c_{\alpha, \|u\|_{L^1(\mathbb{R}^n)}^{\alpha + \beta}.$$ For 1$, any nonnegative extremizer inequality serves best averaging sense above, \neq 1$ corresponds other derivatives. $(n, \beta)=(1,1)$ use Shannon-Whittaker formula prove characteristic $u(x) \chi_{[-1/2,1/2]}$ local among functions defined on $[-1/2,1/2]$ \left\{2,3,4,5,6\right\}$. provide sufficient condition general $\alpha$ terms sign pattern hypergeometric $_1F_2$.