论文标题
有限的汉克尔变换操作员的非偶像行为和频谱的分布
Non-asymptotic behavior and the distribution of the spectrum of the finite Hankel transform operator
论文作者
论文摘要
对于固定的真实，$ c> 0 $，$ a> 0 $和$α> - \ frac {1} {2} $，圆形的pr酸球形波函数（CPSWFS）或2D- slepian函数某些作者称之为它，是$ \ \ nankel $ ns $ ns $ \ \ \ \ \ \;在$ l^2（0,1）$上定义的积分运算符，内核$ h_c^α（x，y）= \ sqrt {cxy}j__α（cxy）$。另外，它们是积极的，自动化的紧凑型积分算子$ \ MATHCAL {q} _C^α= C \ MATHCAL {此外，预计对CREO-EM中的潜在应用程序，对CPSWF而不是傅立叶式贝塞尔的新兴趣有望使其对可转让主要成分分析（PCA）的吸引力。 For this purpose, we give in this paper a precise non-asymptotic estimates for these eigenvalues, within the three main regions of the spectrum of $\mathcal{Q}_c^α$ as well as these distributions in $(0,1).$ Moreover, we describe a series expansion of CPSWFs with respect to the generalized Laguerre functions basis of $L^2(0,\infty)$由$ψ_{n，α}^a（x）= \ sqrt {2} a^{α+1} x^{α+1/2} $ \ widetilde {l} _n^α$是归一化的laguerre多项式。
For a fixed reals $c>0$, $a>0$ and $α>-\frac{1}{2}$, the circular prolate spheroidal wave functions (CPSWFs) or 2d-Slepian functions as some authors call it, are the eigenfunctions of the finite Hankel transform operator, denoted by $\mathcal{H}_c^α$, which is the integral operator defined on $L^2(0,1)$ with kernel $H_c^α(x,y)=\sqrt{cxy}J_α(cxy)$. Also, they are the eigenfunctions of the positive, self-adjoint compact integral operator $\mathcal{Q}_c^α=c\mathcal{H}_c^α\mathcal{H}_c^α.$ The CPSWFs play a central role in many applications such as the analysis of 2d-radial signals. Moreover, a renewed interest on 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 a precise non-asymptotic estimates for these eigenvalues, within the three main regions of the spectrum of $\mathcal{Q}_c^α$ as well as these distributions in $(0,1).$ Moreover, we describe a series expansion of CPSWFs with respect to the generalized Laguerre functions basis of $L^2(0,\infty)$ defined by $ψ_{n,α}^a(x)=\sqrt{2}a^{α+1}x^{α+1/2}e^{-\frac{(ax)^2}{2}}\widetilde{L}_n^α(a^2x^2)$, where $\widetilde{L}_n^α$ is the normalised Laguerre polynomial.
