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: we pick the density $u$ of a probability distribution with mean $0$ and some moment at the desired scale and compute the convolution $u \ast f$. Is there a particularly natural choice for $u$? 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.\[u = \arg \underset{u}{\min} \underset{f \in L^2 (\mathbb{R}^n)}{\sup} \frac{{\lVert \nabla (u \ast f ) \rVert}_{L^2 (\mathbb{R}^n)}}{{\lVert f \rVert}_{L^2 \mathbb{R}^n}} \; \textrm{.}\]This optimal function turns out to be a minimizer of an uncertainty principle: for $\alpha \gt 0$ and $\beta \gt n/2$, there exists $c_{\alpha, \beta, n} \gt 0$ such that for all $u \in L^1(\mathbb{R}^n)$\[{\lVert {\lvert \xi \rvert}^\beta \cdot \hat{u} \rVert}^\alpha_{L^\infty (\mathbb{R}^n)} \cdot {\lVert {\lvert x \rvert}^\alpha \cdot u \rVert}^\beta_{ L^1(\mathbb{R}^n)} \geq c_{\alpha,\beta,n} {\lVert u \rVert}^{\alpha+\beta}_{L^1 (\mathbb{R}^n)} \; \textrm{.}\]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 $\alpha \in \lbrace 2,3,4,5,6 \rbrace$. We provide a sufficient condition for general $\alpha$ in terms of a sign pattern for the hypergeometric function $_1F_2$.

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S.S. is supported by the NSF (DMS-2123224), and by the Alfred P. Sloan Foundation.

Received 5 May 2020

Received revised 26 July 2021

Accepted 15 August 2021

Published 29 August 2022