Computing fourier transforms and convolutions on the 2-sphere

Advances in Applied Mathematics

Volumn 15, Issue 2, 1994, Pages 202-250

  Driscoll, James R ^a   Healy, Dennis M ^a  

^a DARTMOUTH COLLEGE   (United States)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0000273884     PISSN: 01968858     EISSN: 10902074     Source Type: Journal    
DOI: 10.1006/aama.1994.1008     Document Type: Article

References (51)

1. B. Alpert and V. Rokhlin, “A Fast Algorithm for the Evaluation of Legendre Expansions,” Research Report YALEV/DCS/RR-671, Yale University, Department of Computer Science, 1989.
2. A. Aho, J. Hopcroft and J. Ullman, “The Design and Analysis of Computer Algorithms,” Addison Wesley, Reading, MA, 1976.
3. L. Ausländer and R. Tolmieri, Is computing with the Fast Fourier transform pure or applied mathematics?, Bull. Amer. Math. Soc. (N. S.) 1 (1979), 847-897.
4. P. Barrucand and D. Dickinson, On the associated Legendre polynomials, in “Orthogonal Expansions and Their Continuous Analogues,” Southern Illinois University Press, Carbondale, IL, 1968.
5. T. Beth, On the computational complexity of the general discrete Fourier transform, Theoret. Comput. Sei. 51 (1987), 331-339.
6. L. C. Biedenharn and J. D. Louck, “Angular Momentum in Quantum Mechanics,” Addison Wesley, Reading, MA, 1981.
7. R. Bulrisch and J. Stoer, “Introduction to Numerical Analysis,” Springer-Verlag, New York, 1983.
8. D. Calvetti, A stochastic roundoff error analysis for the fast Fourier transform, Math. Comput. 56, No. 194 (1991), 755-774.
9. D. V. S. Chandra, Accumulation of coefficient roundoff error in fast Fourier transforms implemented with logarithmic number system, IEEE Trans. Acoust. Speech Signal Process, 35, No. 11 (1987), 1633-1636.
10. M. Clausen, Spectral transforms for the symmetric group, in “Proc., Third International Workshop on Spectral Techniques, 1988.”
11. J. W. Cooley and J. W. Tukey, An algorithm for machine calculation of complex Fourier series, Math. Comput. 19 (1965), 297-301.
12. H. S. M. Coxeter, “Regular Polytopes,” Dover, New York, 1973.
13. P. Diaconis, Average running time of the fast Fourier transform, J. Algorithms 1 (1980), 187-208.
14. P. Diaconis, “Group Representations in Probability and Statistics,” IMS, Hayward, CA, 1988.
15. P. Diaconis and D. Rockmore, Efficient computation of the Fourier transform on finite groups, J. Amer. Math. Soc. 3, No. 2 (1990), 297-332.
16. G. A. Dilts, Computation of spherical harmonic expansion coefficients via FFTs, J. Comput. Phys., 57, No. 3 (1985), 439-453.
17. W. Freeden, On integral formulas of the (unit) sphere and their application to numerical computation of integrals, Computing 25 (1980), 131-146.
18. F. A. Gilbert, Inverse problems for the Earth’s normal modes, in “Mathematical Problems in the Geophysical Sciences,” Vol. 1, American Mathematical Society, Providence, RI. 1971.
19. N. C. Gallagher, G. L. Wise, and J. W. Allen, A novel approach for the computation of Legendre polynomial expansions, IEEE Trans. Acoust. Speech Signal Process. 26, No. 1 (1978), 105-106.
20. H. Goldstein, “Classical Mechanics,” Addison Wesley, Reading, MA, 1980.
21. G. H. Golub and C. F. Van Loan, “Matrix Computations,” 2nd ed., Johns Hopkins University Press, Baltimore, 1989.
22. M. Hammermesh, “Group Theory,” Addison Wesley, Reading, MA, 1964.
23. D. M. Healy, “A Relationship between Harmonic Analysis on SU(2) and on SL(2, C)/SU(2),” Ph.D. thesis, U.C. San Diego, 1986.
24. S. Helgason, “Groups and Geometric Analysis,” Academic Press, New York, 1984.
25. G. T. Herman (Ed.), “Image Reconstruction from Projections,” Springer-Verlag, New York, 1979.
26. D. V. James, Quantization errors in the fast Fourier transform, IEEE Trans. Acoust. Speech Signal Process. 23, No. 3 (1975), 277-283.
27. T. Kaneko and B. Liu, Accumulation of round-off error in fast Fourier transforms, J. Assoc. Comput. Mach. 17, No. 4 (1970), 637-654.
28. W. Knight and R. Kaiser, A simple fixed-point error bound for the fast Fourier transform, IEEE Trans. Acoust. Speech Signal Process. 27, No. 6 (1979), 615-620.
29. K. Kobayashi, Solution of multi-dimensional neutron transport equation of the spherical harmonics method using the finite Fourier transformation and quadrature formula, Transport Theory Statist. Phys. 14 (1985), 63-81.
30. A. Messiah, “Quantum Mechanics,” Vol. 2, Wiley, New York, 1958.
31. J. Morgenstern, Note on a lower bound of the linear complexity of the fast Fourier transform, J. Assoc. Comput. Mach. 20 (1973), 305-306.
32. S. D. Morgera, Efficient synthesis and implementation of large discrete Fourier transformations, SIAM J. Comput. 9 (1980), 251-272.
33. D. C. Munson, Floating point roundoff error in the fast Fourier transform, IEEE Trans. Acoust. Speech Signal Process. 29, No. 4 (1981), 877-882.
34. W. Neutsch, Optimal spherical designs and numerical integration on the sphere, J. Comput. Phys. 51 (1983), 313-325.
35. A. V. Oppenheim and C. J. Weinstein, Effects of finite register length in digital filtering and the fast Fourier transform, Proc. IEE-E 60, No. 8 (1972), 957-976.
36. S. A. Orzag, Fast eigenfunction transforms, in “Science and Computers,” Academic Press, Orlando, 1986.
37. G. Panda, R. N. Pal, and B. Chatterjee, Error analysis of Good-Winograd algorithm assuming correlated truncation errors, IEEE Trans. Acoust. Speech Signal Process. 31, No. 2 (1983), 508-512.
38. C. H. Papadimitriou, Optimality of the fast Fourier transform, J. Assoc. Comput. Mach. 26 (1979), 95-102.
39. S. Prakash and V. V. Rao, Vector radix FFT error analysis, IEEE Trans. Acoust. Speech Signal Process. 30, No. 5 (1982), 808-811.
40. U. Ramos, Roundoff error analysis of the fast Fourier transform, Math. Comp. 25 (1971), 757-768.
41. R Rockmore, Computation of Fourier transforms on the symmetric group, in “Computers and Mathematics,” Springer-Verlag, New York, 1989.
42. J. T. Schwartz, Mathematics addresses problems in computer vision for advanced robotics, SIAM News 18 (1985), 3.
43. L. Schwartz, “Theorie des Distributions,” Hermann, Paris, 1966.
44. A. Tam, “Optimal Choice of Directions for the Reconstruction of an Object from Its Plane Integrals, Ph.D. thesis, U.C. Berkeley, 1982.
45. A. A. Terras, “Harmonic Analysis on Symmetric Spaces I,” Springer-Verlag, New York, 1985.
46. T Tran-Thong and B. Liu, Fixed point fast Fourier transform error analysis, IEEE Trans. Acoust. Speech Signal Process. 24, No. 6 (1976), 563-573.
47. V Van Loan, “Computational Framework for the Fast Fourier Transform,” SIAM, Philadelphia, 1992.
48. N. J. Vilenkin, “Special Functions and the Theory of Group Representations,” American Mathematical Society, Providence, RI, 1968.
49. J. H. Wilkinson, “Rounding Errors in Algebraic Processes,” Prentice-Hall, Englewood Cliffs, NJ, 1963.
50. J. H. Wilkinson, “The Algebraic Eigenvalue Problem,” Oxford Univ. Press, Oxford, 1988.
51. S. Winograd, On computing the discrete Fourier transform, Math. Comp. 32 (1978), 175-199.