2020/02/01 by Mourad Boulsane, Boulsane, Mourad · 1 citation
Mathematics · Physics and Astronomy · #41A60(primary) #42C10 #Electromagnetic Scattering and Analysis #FOS: Mathematics #Functional Analysis (math.FA) #Mathematical Analysis and Transform Methods #Mathematical functions and polynomials #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.2002.00170
openalex publication_date 2020/02/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
For a fixed reals c>0, a>0 and \α>-\(1)/(2), the circular\nprolate spheroidal wave functions (CPSWFs) or 2d-Slepian functions as some\nauthors call it, are the eigenfunctions of the finite Hankel transform\noperator, denoted by \Hc\α, which is the integral operator\ndefined on L2(0,1) with kernel\nHc\α(x,y)=\√(cxy)J\α(cxy). Also, they are the\neigenfunctions of the positive, self-adjoint compact integral operator\n\Qc\α=c\Hc\α\Hc\α. The\nCPSWFs play a central role in many applications such as the analysis of\n2d-radial signals. Moreover, a renewed interest on the CPSWFs instead of\nFourier-Bessel basis is expected to follow from the potential applications in\nCryo-EM and that makes them attractive for steerable of principal component\nanalysis(PCA). For this purpose, we give in this paper a precise non-asymptotic\nestimates for these eigenvalues, within the three main regions of the spectrum\nof \Qc\α as well as these distributions in (0,1).\nMoreover, we describe a series expansion of CPSWFs with respect to the\ngeneralized Laguerre functions basis of L2(0,\∞) defined by\n\ψn,\αa(x)=\√(2)a\α+1x\α+1/2e-\((ax)2)/(2) widetildeLn\α(a2x2),\nwhere widetildeLn\α is the normalised Laguerre polynomial.\n