$\ell^2$-Linear Independence for the System of Integer Translates of a Square Integrable Function

Linear independence of time frequency translates for special configurations

We prove that for any 4 points in the plane that belong to 2 parallel lines, there is no linear dependence between the associated time-frequency translates of any nontrivial Schwartz function. If mild Diophantine properties are satisfied, we also prove linear independence in the category of $L^2(\R)$ functions.

量子平移的线性无关性

Linear independence of quantum translates

本文研究ℝⁿ上p-Schatten类算子的量子平移线性相关性，明确了不同p值范围下量子平移线性无关与相关的对应判定结论。

最终正函数的时频平移的线性无关性

Linear independence of time-frequency translates for ultimately positive functions

本文研究HRT猜想，证明最终正的平方可积函数与时频平面四点集构成的Gabor系统满足该猜想，并将结果推广到频率在$\mathbb{Z}$上至多有一个线性相关关系的平移集，改进了已有结论。

Oscillation of Fourier transform and Markov-Bernstein inequalities

Under certain conditions on an integrable function f having a real-valued Fourier transform Tf=F, we obtain a certain estimate for the oscillation of F in the interval [-C||f'||/||f||,C||f'||/||f||] with C>0 an absolute constant. Given q>0 and an integrable positive definite function f, satisfying some natural conditions, the above estimate allows us to construct a finite linear combination P of translates f(x+kq)(with k running the integers) such that ||P'||>c||P||/q, where c>0 is another absolute constant. In particular, our construction proves sharpness of an inequality of H. N. Mhaskar for…

A Proof of the HRT Conjecture for Widely Spaced Sets

Given $f \in C_0(\mathbb{R}^n)$ and $Λ\subset \mathbb{R}^{2n}$ a finite set we demonstrate the linear independence of the set of time-frequency translates $\mathcal{G}(f, Λ) = \{π(λ)f\}_{λ\in Λ}$ when the time coordinates of points in $Λ$ are far apart relative to the decay of $f.$ As a corollary, we prove that for any $f \in C_0(\mathbb{R}^n)$ and finite $Λ\subset \mathbb{R}^{2n}$ there exist infinitely many dilations $D_r$ such that $\mathcal{G}(D_rf, Λ)$ is linearly independent. Furthermore, we prove that $\mathcal{G}(f, Λ)$ is linearly independent for functions like $f(t) = \frac{cos(t)}{|…

Refinable shift invariant spaces in R^d

Let $ϕ: \R^d \longrightarrow \C$ be a compactly supported function which satisfies a refinement equation of the form $ϕ(x) = \sum_{k\inΛ} c_k ϕ(Ax - k),\quad c_k\in\C$, where $Γ\subset\R^d$ is a lattice, $Λ$ is a finite subset of $Γ$, and $A$ is a dilation matrix. We prove, under the hypothesis of linear independence of the $Γ$-translates of $ϕ$, that there exists a correspondence between the vectors of the Jordan basis of a finite submatrix of $L=[c_{Ai-j}]_{i,j\inΓ}$ and a finite dimensional subspace $\mathcal H$ in the shift invariant space generated by $ϕ$. We provide a basis of $\mathcal …