关于具有Hermite生成元的Gabor系统的非框架性质及框架集猜想

Sharp results on sampling with derivatives in shift-invariant spaces and multi-window Gabor frames

We study the problem of sampling with derivatives in shift-invariant spaces generated by totally-positive functions of Gaussian type or by the hyperbolic secant. We provide sharp conditions in terms of weighted Beurling densities. As a by-product we derive new results about multi-window Gabor frames with respect to vectors of Hermite functions or totally positive functions.

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…

Multi-window Gabor frames in amalgam spaces

We show that multi-window Gabor frames with windows in the Wiener algebra $W(L^{\infty}, \ell^{1})$ are Banach frames for all Wiener amalgam spaces. As a byproduct of our results we positively answer an open question that was posed by [Krishtal and Okoudjou, Invertibility of the Gabor frame operator on the Wiener amalgam space, J. Approx. Theory, 153(2), 2008] and concerns the continuity of the canonical dual of a Gabor frame with a continuous generator in the Wiener algebra. The proofs are based on a recent version of Wiener's $1/f$ lemma.

拟平移不变空间中的采样与由指数多项式比值生成的Gabor框架

Sampling in quasi shift-invariant spaces and Gabor frames generated by ratios of exponential polynomials

本文引入两类具有周期性的整函数与亚纯函数生成元族，涵盖高斯与双曲正割情形，证明了分离集密度的尖锐采样定理，并应用于半正则格点的Gabor框架密度新结果。

Gabor fusion frames generated by difference sets

Collections of time- and frequency-shifts of suitably chosen generators (Alltop or random vectors) proved successful for many applications in sparse recovery and related fields. It was shown in \cite{xia2005achieving} that taking a characteristic function of a difference set as a generator, and considering only the frequency shifts, gives an equaingular tight frame for the subspace they span. In this paper, we investigate the system of all $N^2$ time- and frequency-shifts of a difference set in dimension $N$ via the mutual coherence, and compare numerically its sparse recovery effectiveness wi…

Wavelet frames, Bergman spaces and Fourier transforms of Laguerre functions

The Fourier transforms of Laguerre functions play the same canonical role in wavelet analysis as do the Hermite functions in Gabor analysis. We will use them as analyzing wavelets in a similar way the Hermite functions were recently by K. Groechenig and Y. Lyubarskii in "Gabor frames with Hermite functions, C. R. Acad. Sci. Paris, Ser. I 344 157-162 (2007)". Building on the work of K. Seip, "Beurling type density theorems in the unit disc, Invent. Math., 113, 21-39 (1993)", concerning sampling sequences on weighted Bergman spaces, we find a sufficient density condition for constructing frames …

From exact systems to Riesz bases in the Balian-Low theorem

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 generator…

Time-frequency shift invariance and the Amalgan Balian Low Theorem

We consider smoothness properties of the generator of a principal Gabor space on the real line which is invariant under some additional translation-modulation pair. We prove that if a Gabor system on a lattice with rational density is a Riesz basis for its closed linear span, and if the closed linear span, a Gabor space, has any additional translation-modulation invariance, then its generator cannot decay well in time and in frequency simultaneously.

无Gabor对偶的Gabor框架

Gabor frames with no Gabor duals

本文证明在任意光滑度下，存在$L^2(\R^d)$上的Gabor框架，其拥有无限多个替代对偶和伪合成对偶，但均非Gabor系统。

Universal Bounds and Monotonicity Properties of Ratios of Hermite and Parabolic Cylinder Functions

We obtain so far unproved properties of a ratio involving a class of Hermite and parabolic cylinder functions. Those ratios are shown to be strictly decreasing and bounded by universal constants. Differently to usual analytic approaches, we employ simple purely probabilistic arguments to derive our results. In particular, we exploit the relation between Hermite and parabolic cylinder functions and the eigenfunctions of the infinitesimal generator of the Ornstein-Uhlenbeck process. As a byproduct, we obtain Turán type inequalities.

Theta 函数的导数及 Lyubarskii 与 Nes 的一个问题

Derivatives of Theta Functions and a Problem of Lyubarskii and Nes

本文刻画了使第一 Hermite 函数生成的 Gabor 系统成为框架的有理密度格，证明其密度满足 D(Λ)=q/p（p,q 互素且 q≥p+2），证实了 Lyubarskii 与 Nes 的猜想，并给出 Lean 4 形式化证明。