ABSTRACT

For fixed reals c>0, a>0 and $\alpha >-\frac{1}{2}$, the circular prolate spheroidal wave functions (CPSWFs) or 2d-Slepian functions are the eigenfunctions of the finite Hankel transform operator, denoted by ${\mathcal{H}}_c^{\alpha }$, which is the integral operator defined on $L^2(0,1)$ with kernel $H_c^{\alpha }(x,y)=\sqrt{cxy}{J}_{\alpha }\left(cxy\right)$. Also, they are the eigenfunctions of the positive, self-adjoint compact integral operator ${\mathcal{Q}}_c^{\alpha }=c{\mathcal{H}}_c^{\alpha }{\mathcal{H}}_c^{\alpha }.$ The CPSWFs play a central role in many applications such as the analysis of 2d-radial signals. Moreover, a renewed interest in the CPSWFs instead of Fourier-Bessel basis is expected to follow from the potential applications in Cryo-EM and that makes them attractive for steerable of principal component analysis(PCA). For this purpose, we give in this paper precise non-asymptotic estimates for the eigenvalues of ${\mathcal{Q}}_c^{\alpha }$, within the three main regions of the spectrum of ${\mathcal{Q}}_c^{\alpha }$. Moreover, we describe a series expansion of CPSWFs with respect to the generalized Laguerre functions basis of $L^2(0,\mathrm{\infty })$ defined by ${\psi }_{n,\alpha }^a\left(x\right)=\sqrt{2}{a}^{\alpha +1}{x}^{\alpha +1/2}{e}^{-\frac{(ax{)}^2}{2}}{\tilde{L}}_n^{\alpha }\left(a^2{x}^2\right)$, where ${\tilde{L}}_n^{\alpha }$ is the normalized Laguerre polynomial.

摘要

对于固定实数c >0, a >0 和$\alpha >-\frac{1}{2}$，圆形长椭球波函数 (CPSWF) 或 2d-Slepian 函数是有限汉克尔变换算子的特征函数，表示为 ${\mathcal{H}}_C^{\alpha }$，这是定义在上的积分运算符 ${升}^2(0,1)$ 带内核 $H_C^{\alpha }(X,是)=\sqrt{CX是}{J}_{\alpha }\left(CX是\right)$. 此外，它们是正的自伴随紧积分算子的本征函数${\mathcal{问}}_C^{\alpha }=C{\mathcal{H}}_C^{\alpha }{\mathcal{H}}_C^{\alpha }.$CPSWF 在许多应用中发挥着核心作用，例如二维径向信号的分析。此外，由于 Cryo-EM 的潜在应用，预计 CPSWF 而不是傅立叶-贝塞尔基将重新引起人们的兴趣，这使得它们对主成分分析 (PCA) 的可控性具有吸引力。为此，我们在本文中给出了特征值的精确非渐近估计${\mathcal{问}}_C^{\alpha }$, 在光谱的三个主要区域内 ${\mathcal{问}}_C^{\alpha }$. 此外，我们描述了关于广义拉盖尔函数基础的 CPSWF 的一系列扩展${升}^2(0,\mathrm{\infty })$ 被定义为 ${\psi }_{n,\alpha }^{\mathrm{一种}}\left(X\right)=\sqrt{2}{\mathrm{一种}}^{\alpha +1}{X}^{\alpha +1/2}{\mathrm{电子}}^{-\frac{(\mathrm{一种}X{)}^2}{2}}{\widetilde{升}}_n^{\alpha }\left({\mathrm{一种}}^2{X}^2\right)$， 在哪里 ${\widetilde{升}}_n^{\alpha }$ 是归一化的拉盖尔多项式。