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.
Similar content being viewed by others
Change history
23 March 2019
Unfortunately Formula 3.6 of the article has been misprinted in original publication and corrected here.
Notes
Our C++ implementation of these fast algorithms is available from https://github.com/cwuelker/SGLPack.
References
Abramowitz, M., Stegun, I.A. (eds.): Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (10th Printing). Department of Commerce, National Bureau of Standards, Gaithersburg, MD, USA (1972)
Andrews, G.E., Askey, R., Roy, R.: Special Functions. Cambridge University Press, Cambridge (1999)
Biedenharn, L.C., Louck, J.D.: Angular Momentum in Quantum Physics: Theory and Application. Addison-Wesley, Boston (1981)
Dai, F., Xu, Y.: Approximation Theory and Harmonic Analysis on Spheres and Balls. Springer, New York (2013)
Erdélyi, A. (ed.): Tables of Integral Transforms, vol. 2. McGraw-Hill, New York (1954)
Freeden, W., Gervens, T., Schreiner, M.: Constructive Approximation on the Sphere: With Applications to Geomathematics. Oxford University Press, Oxford (1998)
Levy, H., Lessman, F.: Finite Difference Equations. Dover, New York (1992)
Nikiforov, A.F., Uvarov, V.B.: Special Functions of Mathematical Physics. Birkhäuser, Basel (1988)
Prestin, J., Wülker, C.: Fast Fourier transforms for spherical Gauss–Laguerre basis functions. In: Pesenson, I., Le Gia, Q.T., Mayeli, A., Mhaskar, H., Zhou, D.-X. (Eds.) Novel Methods in Harmonic Analysis, Vol. 1, Applied and Numerical Harmonic Analysis, pp. 237–263. Birkhäuser, Basel (2017)
Ritchie, D.W.: High-order analytic translation matrix elements for real-space six-dimensional polar Fourier correlations. J. Appl. Cryst. 38(5), 808–818 (2005)
Ritchie, D.W., Kemp, G.J.L.: Protein docking using spherical polar Fourier correlations. Proteins 39(2), 178–194 (2000)
Ritchie, D.W., Kozakov, D., Vajda, S.: Accelerating and focusing protein–protein docking correlations using multi-dimensional rotational FFT generating functions. Bioinformatics 24(17), 1865–1873 (2008)
Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge (1995)
Wülker, C.: Schnelle Fourier-Transformationen für sphärische Gauß-Laguerresche Basisfunktionen. Dissertation, University of Lübeck, Germany (2018)
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.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
The original version of this article was revised: Formula 3.6 has been misprinted in original publication.
Rights and permissions
About this article
Cite this article
Prestin, J., Wülker, C. Translation matrix elements for spherical Gauss–Laguerre basis functions. Int J Geomath 10, 6 (2019). https://doi.org/10.1007/s13137-019-0124-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s13137-019-0124-8
Keywords
- Spherical Gauss–Laguerre (SGL) basis functions
- Translation
- Three-dimensional rigid matching
- Computational harmonic analysis