Young Women in Harmonic Analysis and PDE

December 2-4, 2016

João Pedro Gonçalves Ramos (University of Bonn)

On the equivalence of root uncertainty principles

Let $\widehat{f}(\xi) = \int_{\mathbb{R}} f(x)e^{2 \pi i x \xi} dx$ be the Fourier transform on the real line, and, for every $L^1$, real and even function on the real line, with $f(0),\widehat{f}(0) \le 0$, consider the number $$A(f) := \inf\{ r\ge 0; f(x) \ge 0, \forall x \ge r\}.$$ Inspired by recent bounds on $\mathcal{A} := \inf_{f \text{ as above }} A(f)A(\widehat{f})$ and the classical uncertainty principles in harmonic analysis, we prove the equivalence (in)equalities $$\mathcal{A} = \mathcal{A}_{\mathcal{S}} = \mathcal{A}_{bl} \ge \mathcal{A}_d > 0,476 \mathcal{A} ,$$ where $\mathcal{A}_{\mathcal{S}}$ and $\mathcal{A}_{bl}$ stand for, respectively, the infimum of $A(f)A(\widehat{f})$ taken over the intersection of the function space above with Schwartz functions and functions with compact support, and the last term $\mathcal{A}_d$ is the analogous discrete infimum. That is, if $f:\mathbb{T} \to \mathbb{R}$ is also even and integrable, with $\widehat{f} \in \ell^1(\mathbb{Z})$ and $f(0),\widehat{f}(0) \le 0$, then \begin{equation*} \begin{split} Z(f)& = \inf\{ 1/2 \ge r\ge 0; f(x) \ge 0, \forall x \ge r\}\cr Z(\widehat{f})& = \inf\{ n \ge 0; \widehat{f}(k) \ge 0, \forall k \ge n\}.\cr \end{split} \end{equation*}