Analysis of spectral approximations using prolate spheroidal wave functions

Mathematics of Computation

Volumn 79, Issue 270, 2010, Pages 807-827

  Wang, Li Lian ^a  

^a NANYANG TECHNOLOGICAL UNIVERSITY   (Singapore)

Author keywords

Approximations in sobolev spaces; Bandlimited functions; Prolate spheroidal wave functions; Quasi uniform grids; Spectral methods

Indexed keywords

EID: 77956575471     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-09-02268-6     Document Type: Article

References (34)

1. R. A. Adams. Sobolov Spaces. Academic Press, New York, 1975.
2. C. Bernardi and Y. Maday. Spectral method. In P. G. Ciarlet and L. L. Lions, editors, Handbook of Numerical Analysis, V. 5 (Part 2). North-Holland, 1997.
3. G. Beylkin and K. Sandberg. Wave propagation using bases for bandlimited functions. Wave Motion, 41(3):263-291, 2005.
4. C. J. Bouwkamp. On the theory of spheroidal wave functions of order zero. Nederl. Akad. Wetensch., Proc., 53:931-944 = Indagationes Math. 12, 326-339 (1950), 1950.
5. John P. Boyd. Approximation of an analytic function on a finite real interval by a bandlimited function and conjectures on properties of prolate spheroidal functions. Appl. Comput. Harmon. Anal., 15(2):168-176, 2003.
6. John P. Boyd. Large mode number eigenvalues of the prolate spheroidal differential equation. Appl. Math. Comput., 145(2-3):881-886, 2003.
7. John P. Boyd. Prolate spheroidal wavefunctions as an alternative to Chebyshev and Legendre polynomials for spectral element and pseudospectral algorithms. J. Comput. Phys., 199(2):688-716, 2004.
8. John P. Boyd. Algorithm 840: Computation of grid points, quadrature weights and derivatives for spectral element methods using prolate spheroidal wave functions-prolate elements. ACM Trans. Math. Software, 31(1):149-165, 2005.
9. C. Canuto, M. Y. Hussaini, A. Quarteroni, and T. A. Zang. Spectral methods. Scientific Computation. Springer-Verlag, Berlin, 2006. Fundamentals in single domains.
10. Q.-Y. Chen, D. Gottlieb, and J. S. Hesthaven. Spectral methods based on prolate spheroidal wave functions for hyperbolic PDEs. SIAM J. Numer. Anal., 43(5):1912-1933 (electronic), 2005.
11. Dorota Da̧browska. Recovering signals from inner products involving prolate spheroidals in the presence of jitter. Math. Comp., 74(249):279-290 (electronic), 2005.
12. C. Flammer. Spheroidal Wave Functions. Stanford University Press, Stanford, CA, 1957.
13. Andreas Glaser, Xiangtao Liu, and Vladimir Rokhlin. A fast algorithm for the calculation of the roots of special functions. SIAM J. Sci. Comput., 29(4):1420-1438 (electronic), 2007.
14. Ben-yu Guo and Li-Lian Wang. Jacobi approximations in non-uniformly Jacobi-weighted Sobolev spaces. J. Approx. Theory, 128(1):1-41, 2004.
15. J. Hesthaven, S. Gottlieb, and D. Gottlieb. Spectral Methods for Time-Dependent Problems. Cambridge Monographs on Applied and Computational Mathematics. Cambridge, 2007.
16. Abderrazek Karoui and Tahar Moumni. New efficient methods of computing the prolate spheriodal wave functions and their corresponding eigenvalues. Appl. Comput. Harmon. Anal., 24:269-289, 2008.
17. N. Kovvali, W. Lin, and L. Carin. Pseudospectral method based on prolate spheroidal wave functions for frequency-domain electromagnetic simulations. IEEE Trans. Antennas and Propagation, 53:3990-4000, 2005.
18. Narayan Kovvali, Wenbin Lin, Zhiqin Zhao, Luise Couchman, and Lawrence Carin. Rapid prolate pseudospectral differentiation and interpolation with the fast multipole method. SIAM J. Sci. Comput., 28(2):485-497 (electronic), 2006.
19. H. J. Landau and H. O. Pollak. Prolate spheroidal wave functions, Fourier analysis and uncertainty. III. The dimension of the space of essentially time- and band-limited signals. Bell System Tech. J., 41:1295-1336, 1962.
20. J. Meixner and F. W. Scḧafke. Mathieusche Funktionen und Spḧaroidfunktionen mit Anwendungen auf physikalische und technische Probleme. Springer, Berlin, 1954.
21. Ian C. Moore and Michael Cada. Prolate spheroidal wave functions, an introduction to the Slepian series and its properties. Appl. Comput. Harmon. Anal., 16(3):208-230, 2004.
22. Vladimir Rokhlin and Hong Xiao. Approximate formulae for certain prolate spheroidal wave functions valid for large values of both order and band-limit. Appl. Comput. Harmon. Anal., 22(1):105-123, 2007.
23. Jie Shen. Efficient spectral-Galerkin method I. Direct solvers for second- and fourth-order equations using Legendre polynomials. SIAM J. Sci. Comput., 15:1489-1505, 1994.
24. Yoel Shkolnisky, Mark Tygert, and Vladimir Rokhlin. Approximation of bandlimited functions. Appl. Comput. Harmon. Anal., 21(3):413-420, 2006.
25. D. Slepian and H. O. Pollak. Prolate spheroidal wave functions, Fourier analysis and uncertainty. I. Bell System Tech. J., 40:43-63, 1961.
26. David Slepian. Prolate spheroidal wave functions, Fourier analysis and uncertainity. IV. Extensions to many dimensions; generalized prolate spheroidal functions. Bell System Tech. J., 43:3009-3057, 1964.
27. Mark A. Taylor and Beth A. Wingate. A generalization of prolate spheroidal functions with more uniform resolution to the triangle. J. Engrg. Math., 56(3):221-235, 2006.
28. Roger Temam. Infinite-dimensional dynamical systems in mechanics and physics, volume 68 of Applied Mathematical Sciences. Springer-Verlag, New York, 1988.
29. G. Walter and T. Soleski. A new friendly method of computing prolate spheroidal wave functions and wavelets. Appl. Comput. Harmon. Anal., 19(3):432-443, 2005.
30. Gilbert G. Walter. Prolate spheroidal wavelets: Translation, convolution, and differentiation made easy. J. Fourier Anal. Appl., 11(1):73-84, 2005.
31. Gilbert G. Walter and Xiaoping Shen. Wavelets based on prolate spheroidal wave functions. J. Fourier Anal. Appl., 10(1):1-26, 2004.
32. H. Xiao. Prolate spheroidal wave functions, quadrature, interpolation, and asymptotic formulae. Ph.D. Thesis, Yale University, 2001.
33. H. Xiao, V. Rokhlin, and N. Yarvin. Prolate spheroidal wavefunctions, quadrature and interpolation. Inverse Problems, 17(4):805-838, 2001. Special issue to celebrate Pierre Sabatier's 65th birthday (Montpellier, 2000).
34. Ahmed I. Zayed. A generalization of the prolate spheroidal wave functions. Proc. Amer. Math. Soc., 135(7):2193-2203 (electronic), 2007.