Shannon Multiresolution Analysis on the Heisenberg Group

各向异性 Rényi 熵的最优上界与海森堡不确定性原理

Sharp upper bound for anisotropic Rényi entropy and Heisenberg uncertainty principle

本文在 Folland-Stein 齐次李群上证明了具有最佳常数的 Renyi 熵各向异性 Shannon 不等式及最优 Shannon 不等式，并利用分层群的对数 Sobolev 不等式证明了海森堡型不确定性原理。

Reconstruction of Paley-Wiener functions on the Heisenberg group

Let $M$ be a Riemmanian manifold with bounded geometry. We consider a generalization of Paley-Wiener functions and Lagrangian splines on $M$. An analog of the Paley-Wiener theorem is given. We also show that every Paley-Wiener function on a manifold is uniquely determined by its values on some discrete sets of points. The main result of the paper is a generalization of the Whittaker-Shannon formula for reconstruction of a Paley-Wiener function from its values on a discrete set. It is shown that every Paley- Wiener function on $M$ is a limit of some linear combinations of fundamental solutions …

Gabor fields and wavelet sets for the Heisenberg group

We study singly-generated wavelet systems on $\Bbb R^2$ that are naturally associated with rank-one wavelet systems on the Heisenberg group $N$. We prove a necessary condition on the generator in order that any such system be a Parseval frame. Given a suitable subset $I$ of the dual of $N$, we give an explicit construction for Parseval frame wavelets that are associated with $I$. We say that $g\in L^2(I\times \Bbb R)$ is Gabor field over $I$ if, for a.e. $λ\in I$, $|λ|^{1/2} g(λ,\cdot)$ is the Gabor generator of a Parseval frame for $L^2(\Bbb R)$, and that $I$ is a Heisenberg wavelet set if ev…