论文标题
在指数基础和具有非线性相函数的框架和某些应用
On exponential bases and frames with non-linear phase functions and some applications
论文作者
论文摘要
在本文中，我们研究了$ e（λ，φ）= \ {e^{2πiλ\cdotφ（x）}类型的指数系统的光谱和框架 - 光谱：λ\inλ\} $，其中相位函数$φ$是不必固定的孔函数。对$ e（λ，φ）$是$ l^{2}（μ）$的$ e（λ，φ）$的$（λ，φ）$的完整表征。特别是，我们表明，中间三分之一的第托措施和单位光盘，每个圆盘都以一定的非线性相位接受正交基础。在相位函数上的自然规律性条件下，当$μ$是$ [0,1] $和$λ= {\ mathbb {z}}}上的lebesgue度量时，我们只表明只有标准相位函数$φ（x）= \ pm x $是唯一引起正常基碱的可能功能。令人惊讶的是，但是我们证明，即使在更高维度中的连续可区分相位函数中，也存在更大程度的灵活性。例如，我们能够描述在$ {\ mathbb {r}}}^{d} $上定义的大量功能$φ$，这样系统$ e（λ，φ）$是$ l^{2} [2} [0,1]^d} $ d d \ divity的班级的正常基础。局部紧凑型基团，用于建造正统基础。最后，我们通过说出几个开放问题来结束论文。
In this paper, we study the spectrality and frame-spectrality of exponential systems of the type $E(Λ,φ) = \{e^{2πi λ\cdotφ(x)}: λ\inΛ\}$ where the phase function $φ$ is a Borel measurable which is not necessarily linear. A complete characterization of pairs $(Λ,φ)$ for which $E(Λ,φ)$ is an orthogonal basis or a frame for $L^{2}(μ)$ is obtained. In particular, we show that the middle-third Cantor measures and the unit disc, each admits an orthogonal basis with a certain non-linear phase. Under a natural regularity condition on the phase functions, when $μ$ is the Lebesgue measure on $[0,1]$ and $Λ= {\mathbb{Z}},$ we show that only the standard phase functions $φ(x) = \pm x$ are the only possible functions that give rise to orthonormal bases. Surprisingly, however we prove that there exist a greater degree of flexibility, even for continuously differentiable phase functions in higher dimensions. For instance, we were able to describe a large class of functions $φ$ defined on ${\mathbb{R}}^{d}$ such that the system $E(Λ,φ)$ is an orthonormal basis for $L^{2}[0,1]^{d}$ when $d\geq2.$ Moreover, we discuss how our results apply to the discretization problem of unitary representations of locally compact groups for the construction of orthonormal bases. Finally, we conclude the paper by stating several open problems.
