Translation matrix elements for spherical Gauss–Laguerre basis functions

  • Jürgen Prestin
  • Christian WülkerEmail author
Original Paper
Part of the following topical collections:
  1. Mathematical Problems in Medical Imaging and Earth Sciences


Spherical Gauss–Laguerre (SGL) basis functions, i.e., normalized functions of the type \(L_{n-l-1}^{(l + 1/2)}(r^2) r^{l} Y_{lm}(\vartheta ,\varphi ), |m| \le l < n \in {\mathbb {N}}\), constitute an orthonormal polynomial basis of the space \(L^{2}\) on \({\mathbb {R}}^{3}\) with radial Gaussian weight \(\exp (-r^{2})\). We have recently described reliable fast Fourier transforms for the SGL basis functions. The main application of the SGL basis functions and our fast algorithms is in solving certain three-dimensional rigid matching problems, where the center is prioritized over the periphery. For this purpose, so-called SGL translation matrix elements are required, which describe the spectral behavior of the SGL basis functions under translations. In this paper, we derive a closed-form expression of these translation matrix elements, allowing for a direct computation of these quantities in practice.


Spherical Gauss–Laguerre (SGL)basis functions Translation Three-dimensional rigid matching Computational harmonic analysis 

Mathematics Subject Classification

65D20 33F05 



The authors would like to thank the referees for their valuable comments. Furthermore, the authors are grateful to Sabrina Kombrink, Vitalii Myroniuk, and Nadiia Derevianko for scientific discussion and helpful comments on the manuscript.


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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institute of MathematicsUniversity of LübeckLübeckGermany
  2. 2.Department of Mechanical EngineeringJohns Hopkins UniversityBaltimoreUSA

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