相关论文: Uncertainty Principles for Compact Groups

Let $G$ be a locally compact abelian group, and let $\widehat{G}$ denote its dual group, equipped with a Haar measure. A variant of the uncertainty principle states that for any $S \subset G$ and $\Sigma \subset \widehat{G}$, there exists a…

Let $G$ be a finite abelian group, and let $f: G \to \C$ be a complex function on $G$. The uncertainty principle asserts that the support $\supp(f) := \{x \in G: f(x) \neq 0\}$ is related to the support of the Fourier transform $\hat f: G…

The aim of this paper is to prove new uncertainty principles for an integral operator $\tt$ with a bounded kernel for which there is a Plancherel theorem. The first of these results is an extension of Faris's local uncertainty principle…

A well-known version of the uncertainty principle on the cyclic group $\mathbb{Z}_N$ states that for any couple of functions $f,g\in\ell^2(\mathbb{Z}_N)\setminus\{0\}$, the short-time Fourier transform $V_g f$ has support of cardinality at…

We present a sufficient condition on sets $E$ and $F$ in $\mathbb{R}^d$ to ensure compactness of Fourier concentration operators by introducing the notion of sets which are very thin at infinity. We are able to show that if the sets $E$ and…

For unbounded operators A,B and C in general, with C closure of [A,B] does not lead to the uncertainty relation ||Au|| ||Bu|| >= |<C u,u> |/2. If A,B and C are part of the generators of a unitary representation of a Lie group then the…

Helgason showed that a given measure $f\in M(G)$ on a compact group $G$ should be in $L^2(G)$ automatically if all random Fourier series of $f$ are in $M(G)$. We explore a natural analogue of the theorem in the framework of compact quantum…

We define lacunary Fourier series on a compact connected semisimple Lie group $G$. If $f \in L^1(G)$ has lacunary Fourier series, and vanishes on a non empty open set, then we prove that $f$ vanishes identically. This may be viewed as a…

Let G be a locally compact Hausdorff group in which every element is of finite order, and let P(G) denote the class of all regular probability measures on G. In this note, it is observed that a characterization of algebraically regular…

We calculate the norm of the Fourier operator from $L^p(X)$ to $L^q(\hat{X})$ when $X$ is an infinite locally compact abelian group that is, furthermore, compact or discrete. This subsumes the sharp Hausdorff-Young inequality on such…

We prove various Hardy-type and uncertainty inequalities on a stratified Lie group $G$. In particular, we show that the operators $T_\alpha: f \mapsto |.|^{-\alpha} L^{-\alpha/2} f$, where $|.|$ is a homogeneous norm, $0 < \alpha < Q/p$,…

We extend a classical theorem of Courr\`{e}ge to Lie groups in a global setting, thus characterising all linear operators on the space of smooth functions of compact support that satisfy the positive maximum principle. We show that these…

If $G$ is a compact Lie group endowed with a left invariant metric $g$, then $G$ acts via pullback by isometries on each eigenspace of the associated Laplace operator $\Delta_g$. We establish algebraic criteria for the existence of left…

We prove a new uncertainty principle for square-integrable irreducible unitary representations of connected Lie groups. The concentration of the matrix coefficients is measured in terms of weighted $L^p$ norms, with weights in the local…

For a family of weight functions $h_\kappa$ that are invariant under a reflection group, the uncertainty principle on the unit sphere in the form of $$ \min_{1 \le i \le d} \int_{\mathbb{S}^{d-1}} (1- x_i) |f(x)|^2 h_\kappa^2(x) d\sigma…

Let G be a finite abelian group of order n. For a complex valued function f on G, let \fht denote the Fourier transform of f. The uncertainty inequality asserts that if f \neq 0 then |supp(f)| |supp(\fht)| \geq n. Answering a question of…

Extending a recent result by Frank and Lieb, we show an entropic uncertainty principle for mixed states in a Hilbert space relatively to pairs of positive operator valued measures that are independent in some sense. This yields…

For any finite group $G$, any transitive $G$-set $X$ and any field ${\Bbb F}$, we consider the vector space ${\Bbb F}^X$ of all functions from $X$ to ${\Bbb F}$, which is a $G$-space isomorphic to the permutation ${\Bbb F} G$-module ${\Bbb…

In this work, we point out an overlooked and subtle feature of the generalized uncertainty principle (GUP) approach to quantizing gravity: namely that different pairs of modified operators with the same modified commutator,…

We introduce a continuous analog of the Fourier ratio for compactly supported Borel measures. For a measure \(\mu\) on \(\mathbb{R}^d\) and \(f\in L^2(\mu)\), the Fourier ratio compares \(L^1\) and \(L^2\) norms of a regularized Fourier…