Translation matrix elements for spherical Gauss–Laguerre basis functions
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Abstract
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.
Keywords
Spherical Gauss–Laguerre (SGL)basis functions Translation Three-dimensional rigid matching Computational harmonic analysisMathematics Subject Classification
65D20 33F05Notes
Acknowledgements
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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