Uncertainty principles for orthonormal bases

Non-symmetric Convex Polytope and Gabor Orthonormal Bases

In this paper, we show that the non-symmetric convex polytope cannot be served as a window function to produce a Gabor orthonormal basis by any time-frequency sets.

紧流形上的不确定性原理、谱局部化和奇异薛定谔算子

Uncertainty Principles, Spectral Localization, and Singular Schrödinger Operators on Compact Manifolds

通过结合正交系统的限制估计与拉普拉斯-贝尔特拉米和薛定谔算子的谱投影界，建立了紧无边黎曼流形上的不确定性原理，并应用于不完全谱数据恢复的唯一性结果。

Hardy space theory on spaces of homogeneous type via orthonormal wavelet bases

In this paper, using the remarkable orthonormal wavelet basis constructed recently by Auscher and Hytönen, we establish the theory of product Hardy spaces on spaces ${\widetilde X} = X_1\times X_2\times\cdot \cdot\cdot\times X_n$, where each factor $X_i$ is a space of homogeneous type in the sense of Coifman and Weiss. The main tool we develop is the Littlewood--Paley theory on $\widetilde X$, which in turn is a consequence of a corresponding theory on each factor space. We define the square function for this theory in terms of the wavelet coefficients. The Hardy space theory developed in this…

Gabor 标准正交基、铺砌与周期性

Gabor orthonormal bases, tiling and periodicity

本文证明紧支撑窗函数的 Gabor 系统若构成 L²(R) 标准正交基，则其时频平移集合均为周期集，并借助一个具有独立意义的泛函铺砌型必要条件完成论证。

复球上齐次多项式的一致有界标准正交基

Uniformly Bounded Orthonormal Bases of Homogeneous Polynomials on the Complex Sphere

该研究解决了复球上齐次多项式的一致有界标准正交基的Rudin长期问题，证明了对任意$K \geq 2$和$N \geq 0$存在满足条件的基，填补了$K \geq 4$情况的空白。

On Gabor orthonormal bases over finite prime fields

We study Gabor orthonormal windows in $L^2({\Bbb Z}_p^d)$ for translation and modulation sets $A$ and $B$, respectively, where $p$ is prime and $d\geq 2$. We prove that for a set $E\subset \Bbb Z_p^d$, the indicator function $1_E$ is a Gabor window if and only if $E$ tiles and is spectral. Moreover, we prove that for any function $g:\Bbb Z_p^d\to \Bbb C$ with support $E$, if the size of $E$ coincides with the size of the modulation set $B$ or if $g$ is positive, then $g$ is a unimodular function, i.e., $|g|=c1_E$, for some constant $c>0$, and $E$ tiles and is spectral. We also prove the existe…

Orthonormal bases of regular wavelets in spaces of homogeneous type

Adapting the recently developed randomized dyadic structures, we introduce the notion of spline function in geometrically doubling quasi-metric spaces. Such functions have interpolation and reproducing properties as the linear splines in Euclidean spaces. They also have Hölder regularity. This is used to build an orthonormal basis of Hölder-continuous wavelets with exponential decay in any space of homogeneous type. As in the classical theory, wavelet bases provide a universal Calderón reproducing formula to study and develop function space theory and singular integrals. We discuss the example…

Non-separable lattices, Gabor orthonormal bases and Tilings

Let $K\subset \Bbb R^d$ be a set with positive and finite Lebesgue measure. Let $Λ=M(\Bbb Z^{2d})$ be a lattice in $\Bbb R^{2d}$ with density dens$(Λ)=1$. It is well-known that if $M$ is a diagonal block matrix with diagonal matrices $A$ and $B$, then $\mathcal G(|K|^{-1/2}χ_K, Λ)$ is an orthonormal basis for $L^2(\Bbb R^d)$ if and only if $K$ tiles both by $A(\Bbb Z^d)$ and $B^{-t}(\Bbb Z^d)$. However, there has not been any intensive study when $M$ is not a diagonal matrix. We investigate this problem for a large class of important cases of $M$. In particular, if $M$ is any lower block trian…

平移不变核的正交展开

Orthonormal Expansions for Translation-Invariant Kernels

本文提出一种基于傅里叶分析的通用方法，从 $\mathscr{L}_2(\mathbb{R})$ 的正交基出发构造平移不变核的正交展开，并给出 Matérn、Cauchy 和高斯核的显式展开。

On the spectra of a Cantor measure

We analyze all orthonormal bases of exponentials on the Cantor set defined by Jorgensen and Pedersen in J. Anal. Math. 75,1998, pp 185-228. A complete characterization for all maximal sets of orthogonal exponentials is obtained by establishing a one-to-one correspondence with the spectral labelings of the infinite binary tree. With the help of this characterization we obtain a sufficient condition for a spectral labeling to generate a spectrum (an orthonormal basis). This result not only provides us an easy and efficient way to construct various of new spectra for the Cantor measure but also e…

Spectral properties of some unions of linear spaces

We consider \textit{additive spaces}, consisting of two intervals of unit length or two general probability measures on ${\mathbb R}^1$, positioned on the axes in ${\mathbb R}^2$, with a natural additive measure $ρ$. We study the relationship between the exponential frames, Riesz bases, and orthonormal bases of $L^2(ρ)$ and those of its component spaces. We find that the existence of exponential bases depends strongly on how we position our measures on ${\mathbb R}^1$. We show that non-overlapping additive spaces possess Riesz bases, and we give a necessary condition for overlapping spaces. We…

Sampling and Frequency Warping

Optimal sampling of non band-limited functions is an issue of great importance that has attracted considerable attention. We propose to tackle this problem through the use of a frequency warping: First, by a nonlinear shrinking of frequencies, the function is transformed into a band-limited one. One may then perform a decomposition in Fourier series. This process gives rise to new orthonormal bases of the Sobolev spaces H^alpha. When alpha is an integer, these orthonormal bases can be expressed in terms of Laguerre functions. We study the reconstruction and speed of convergence properties of t…