深度零问题与HRT猜想

Proof of the HRT conjecture for almost every (1,3) configuration

Based on the proof of Demeter \We prove that for almost every (1,3) configuration, there is no linear dependence between the associated time-frequency translates of any $f\in L^2(\R)\backslash \{0\}$.cite{Dem10}, we establish the HRT conjecture for almost every (1,3) configuration.

A note on the HRT conjecture and a new uncertainty principle for the short-time Fourier transform

In this note we provide a negative answer to a question raised by M. Kreisel concerning a condition on the short-time Fourier transform that would imply the HRT conjecture. In particular we provide a new type of uncertainty principle for the short-time Fourier transform which forbids the arrangement of an arbitrary "bump with fat tail" profile.

Zak变换的零点集与HRT猜想

Zero set of Zak transform and the HRT Conjecture

本文围绕HRT猜想，探讨了向量$f$的Zak变换零点集，以及时频平面上整数格中排除特定点的$N$点排布，为研究时频平移向量的线性无关性提供了新发现。

两类特殊构型的HRT猜想

The HRT conjecture for two classes of special configurations

本文利用遍历定理证明了HRT猜想在两类特殊构型下成立：一类是N个点位于整数格、另一点任意；另一类是N个点共线、另一点不在该线上，从而彻底解决了后者。

Interpolation in Wavelet Spaces and the HRT-Conjecture

We investigate the wavelet spaces $\mathcal{W}_{g}(\mathcal{H}_π)\subset L^{2}(G)$ arising from square integrable representations $π:G \to \mathcal{U}(\mathcal{H}_π)$ of a locally compact group $G$. We show that the wavelet spaces are rigid in the sense that non-trivial intersection between them imposes strong conditions. Moreover, we use this to derive consequences for wavelet transforms related to convexity and functions of positive type. Motivated by the reproducing kernel Hilbert space structure of wavelet spaces we examine an interpolation problem. In the setting of time-frequency analysi…

对称构型与实值函数的HRT猜想

The HRT Conjecture for Symmetric Configurations and Real-Valued Functions

本文针对无穷族对称(2n+1,2)构型及实值函数，结合三角多项式乘积归约与无理旋转轨道估计，证明了HRT猜想，且涵盖所有含四个不同点的构型。