Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Mathematics of Computation
Mathematics of Computation
ISSN 1088-6842(online) ISSN 0025-5718(print)

 

The resolution of the Gibbs phenomenon
for spherical harmonics


Author: Anne Gelb
Journal: Math. Comp. 66 (1997), 699-717
MSC (1991): Primary 42A10, 41A10, 41A25
MathSciNet review: 1401940
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: Spherical harmonics have been important tools for solving geophysical and astrophysical problems. Methods have been developed to effectively implement spherical harmonic expansion approximations. However, the Gibbs phenomenon was already observed by Weyl for spherical harmonic expansion approximations to functions with discontinuities, causing undesirable oscillations over the entire sphere.

Recently, methods for removing the Gibbs phenomenon for one-dimensional discontinuous functions have been successfully developed by Gottlieb and Shu. They proved that the knowledge of the first $N$ expansion coefficients (either Fourier or Gegenbauer) of a piecewise analytic function $f(x)$ is enough to recover an exponentially convergent approximation to the point values of $f(x)$ in any subinterval in which the function is analytic.

Here we take a similar approach, proving that knowledge of the first $N$ spherical harmonic coefficients yield an exponentially convergent approximation to a spherical piecewise smooth function $f(\theta ,\phi )$ in any subinterval $[\theta _1,\theta _2], \break \phi \in [0,2\pi ]$, where the function is analytic. Thus we entirely overcome the Gibbs phenomenon.


References [Enhancements On Off] (What's this?)

  • 1. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, 1970.
  • 2. Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger, and Francesco G. Tricomi, Higher transcendental functions. Vol. II, Robert E. Krieger Publishing Co. Inc., Melbourne, Fla., 1981. Based on notes left by Harry Bateman; Reprint of the 1953 original. MR 698780 (84h:33001b)
  • 3. E. Butkov, Mathematical Physics, Addison-Wesley Publishing Company, 1968.
  • 4. David Gottlieb and Steven A. Orszag, Numerical analysis of spectral methods: theory and applications, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1977. CBMS-NSF Regional Conference Series in Applied Mathematics, No. 26. MR 0520152 (58 #24983)
  • 5. David Gottlieb, Chi-Wang Shu, Alex Solomonoff, and Hervé Vandeven, On the Gibbs phenomenon. I. Recovering exponential accuracy from the Fourier partial sum of a nonperiodic analytic function, J. Comput. Appl. Math. 43 (1992), no. 1-2, 81–98. Orthogonal polynomials and numerical methods. MR 1193295 (94h:42006), http://dx.doi.org/10.1016/0377-0427(92)90260-5
  • 6. D. Gottlieb and C.-W. Shu, On The Gibbs Phenomenon III: recovering exponential accuracy in a sub-interval from the spectral partial sum of a piecewise analytic function, SIAM J. Numer. Anal., 33:1 (1996), 280-290. CMP 96:09
  • 7. D. Gottlieb and C.-W. Shu, On The Gibbs Phenomenon IV: recovering exponential accuracy in a sub-interval from a Gegenbauer partial sum of a piecewise analytic function, Math. Comp. 64:211 (1995), 1081-1095. CMP 95:11
  • 8. I. S. Gradshteyn and I. M. Ryzhik, Table of integrals, series, and products, Academic Press [Harcourt Brace Jovanovich Publishers], New York, 1980. Corrected and enlarged edition edited by Alan Jeffrey; Incorporating the fourth edition edited by Yu. V. Geronimus [Yu. V. Geronimus] and M. Yu. Tseytlin [M. Yu. Tseĭtlin]; Translated from the Russian. MR 582453 (81g:33001)
  • 9. Fritz John, Partial differential equations, 4th ed., Applied Mathematical Sciences, vol. 1, Springer-Verlag, New York, 1982. MR 831655 (87g:35002)
  • 10. S. Orszag, Fourier Series on Spheres, Mon. Wea. Rev. 102 (1978), 56-75.
  • 11. Paul N. Swarztrauber, On the spectral approximation of discrete scalar and vector functions on the sphere, SIAM J. Numer. Anal. 16 (1979), no. 6, 934–949. MR 551317 (81c:65011), http://dx.doi.org/10.1137/0716069
  • 12. H. Weyl, Die Gibbssche Erscheinung in der Theorie der Kugelfunktionen, , Springer-Verlag, 1968, 305-320.

Similar Articles

Retrieve articles in Mathematics of Computation of the American Mathematical Society with MSC (1991): 42A10, 41A10, 41A25

Retrieve articles in all journals with MSC (1991): 42A10, 41A10, 41A25


Additional Information

Anne Gelb
Affiliation: Division of Applied Mathematics, Brown University, Providence, Rhode Island 02912
Email: ag@cfm.brown.edu

DOI: http://dx.doi.org/10.1090/S0025-5718-97-00828-4
PII: S 0025-5718(97)00828-4
Keywords: Gibbs phenomenon, Gegenbauer polynomials, spherical harmonics, exponential accuracy
Received by editor(s): October 26, 1995
Received by editor(s) in revised form: May 1, 1996
Additional Notes: This work was supported in part by AFOSR grant F49620-95-1-0074, NSF grant DMS 95008 14, and DARPA/ONR AASERT grant N00014-93-1-0985.
Article copyright: © Copyright 1997 American Mathematical Society