From exact systems to Riesz bases in the Balian-Low theorem
Shahaf Nitzan, Jan-Fredrik Olsen
备注：
v1: 16 pages. v2: 35 pages. Reorganized the presentation, corrected some typos, and added some additional results. In particular, added proof that main theorem is sharp
We look at the time-frequency localisation of generators of lattice Gabor systems. For a generator of a Riesz basis, this localisation is described by the classical Balian-Low theorem. We establish Balian-Low type theorems for complete and minimal Gabor systems with a frame-type approximation property. These results describe how the best possible localisation of a generator is limited by the degree of control over the coefficients in approximations given by the system, and provide a continuous transition between the classical Balian-Low conditions and the corresponding conditions for generators of complete and minimal systems. Moreover, this holds for the non-symmetric generalisations of these theorems as well.
Multi-tiling sets, Riesz bases, and sampling near the critical density in LCA groups
本文证明了局部紧交换群G的Paley-Wiener空间在临界密度附近的采样集和插值集的存在性。这解决了Gröchenig，Kutyniok和Seip在文章中留下的问题：“LCA群的Landau密度条件”（J. of Funct. Anal. 255（2008）1831-1850）。为了实现这一结果，我们证明了存在的普遍Riesz基地的特点，L2（Ω），提供了相对紧凑的子集Ω的对偶群G满足一个multi-tiling条件。这最后一个结果推广Fuglede定理，并延伸到LCA组设置最近的建设Riesz基地的指数在有界集的Rd。
We prove an almost everywhere convergence result for Bochner-Riesz means of $L^p$ functions on Heisenberg-type groups, yielding the existence of a $p>2$ for which convergence holds for means of arbitrarily small order. The proof hinges on a reduction of weighted $L^2$ estimates for the maximal Bochner-Riesz operator to corresponding estimates for the non-maximal operator, and a `dual Sobolev trace lemma', whose proof is based on refined estimates for Jacobi polynomials.
We prove that if $I_\ell = [a_\ell,b_\ell)$, $\ell=1, \ldots, L$, are disjoint intervals in $[0,1)$ with the property that the numbers $1, a_1, \ldots, a_L, b_1, \ldots, b_L$ are linearly independent over $\mathbb{Q}$, then there exist pairwise disjoint sets $Λ_\ell \subset \mathbb{Z}$, $\ell=1, \ldots, L$, such that for every $J \subset \{ 1, \ldots , L \}$, the system $\{e^{2πi λx} : λ\in \cup_{\ell \in J} \, Λ_\ell \}$ is a Riesz basis for $L^2 ( \cup_{\ell \in J} \, I_\ell)$. Also, we show that for any disjoint intervals $I_\ell$, $\ell=1, \ldots, L$, contained in $[1,N)$ with $N \in \math…
We measure a sign interlacing phenomenon for Bessel sequences $ (u_{k})$ in $ L^{2}$ spaces in terms of the Kantorovich--Rubinstein mass moving norm $ \Vert u_{k}\Vert_{KR}$. Our main observation shows that, quantitatively, the rate of the decreasing $ \Vert u_{k}\Vert_{KR}\longrightarrow 0$ havily depends on S. Bernstein $ n$-widths of a compact of Lipschitz functions. In particular, it depends on the dimension of the measure space. We have sharp results on the worst and the best rate of convergence of Kantorovich--Rubinstein norms of frames on $d$-dimensional cube. Those rates are sharp.
It is known that if $Ω\subset \mathbb{R}^{d}$ belongs to a class of multi-tiling domains when translated by a lattice $Λ$, there exists a Riesz basis of exponentials for $L^{2}(Ω)$ constructed using $k$ translates of the dual lattice $Λ^*$. In this paper, we give an explicit construction of the corresponding biorthogonal dual Riesz basis. We also extend the iterative reconstruction algorithm introduced in prior work to this setting.
Despite the recent advances in the theory of exponential Riesz bases, it is yet unknown whether there exists a set $S \subset \mathbb{R}^d$ which does not admit a Riesz spectrum, meaning that for every $Λ\subset \mathbb{R}^d$ the set of exponentials $e^{2πi λ\cdot x}$ with $λ\inΛ$ is not a Riesz basis for $L^2(S)$. As a meaningful step towards finding such a set, we construct a set $S \subset [-\frac{1}{2}, \frac{1}{2}]$ which does not admit a Riesz spectrum containing a nonempty periodic set with period belonging in $α\mathbb{Q}_+$ for any fixed constant $α> 0$, where $\mathbb{Q}_+$ denotes t…
Motivated by the open problem of exhibiting a subset of Euclidean space which has no exponential Riesz basis, we focus on exponential Riesz bases in finite abelian groups. We point out that that every subset of a finite abelian group has such a basis, removing interest in the existence question in this context. We then define tightness quantities for subsets to measure the conditioning of Riesz bases; for normalized tightness quantities, a value of one corresponds to an orthogonal basis, and a value of infinity corresponds to nonexistence of a basis. As an application, we obtain new weak evide…
