The resolution of the Gibbs phenomenon for spherical harmonics
Author:
Anne Gelb
Journal:
Math. Comp. 66 (1997), 699717
MSC (1991):
Primary 42A10, 41A10, 41A25
MathSciNet review:
1401940
Fulltext 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 onedimensional discontinuous functions have been successfully developed by Gottlieb and Shu. They proved that the knowledge of the first expansion coefficients (either Fourier or Gegenbauer) of a piecewise analytic function is enough to recover an exponentially convergent approximation to the point values of in any subinterval in which the function is analytic. Here we take a similar approach, proving that knowledge of the first spherical harmonic coefficients yield an exponentially convergent approximation to a spherical piecewise smooth function in any subinterval , where the function is analytic. Thus we entirely overcome the Gibbs phenomenon.
 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, AddisonWesley 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. CBMSNSF Regional Conference Series in Applied
Mathematics, No. 26. MR 0520152
(58 #24983)
 5.
David
Gottlieb, ChiWang
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. 12, 81–98. Orthogonal polynomials and numerical methods. MR 1193295
(94h:42006), http://dx.doi.org/10.1016/03770427(92)902605
 6.
D. Gottlieb and C.W. Shu, On The Gibbs Phenomenon III: recovering exponential accuracy in a subinterval from the spectral partial sum of a piecewise analytic function, SIAM J. Numer. Anal., 33:1 (1996), 280290. CMP 96:09
 7.
D. Gottlieb and C.W. Shu, On The Gibbs Phenomenon IV: recovering exponential accuracy in a subinterval from a Gegenbauer partial sum of a piecewise analytic function, Math. Comp. 64:211 (1995), 10811095. 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, SpringerVerlag, New York, 1982. MR 831655
(87g:35002)
 10.
S. Orszag, Fourier Series on Spheres, Mon. Wea. Rev. 102 (1978), 5675.
 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, , SpringerVerlag, 1968, 305320.
 1.
 M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, 1970.
 2.
 H. Bateman, Higher Transcendental Functions, v2, McGrawHill, 1953. MR 84h:33001b
 3.
 E. Butkov, Mathematical Physics, AddisonWesley Publishing Company, 1968.
 4.
 D. Gottlieb and S. Orszag, Numerical Analysis of Spectral Methods: Theory and Applications, SIAMCBMS, Philadelphia, 1977. MR 58:24983
 5.
 D. Gottlieb, C.W. Shu, A. Solomonoff and H. 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), 8192. MR 94h:42006
 6.
 D. Gottlieb and C.W. Shu, On The Gibbs Phenomenon III: recovering exponential accuracy in a subinterval from the spectral partial sum of a piecewise analytic function, SIAM J. Numer. Anal., 33:1 (1996), 280290. CMP 96:09
 7.
 D. Gottlieb and C.W. Shu, On The Gibbs Phenomenon IV: recovering exponential accuracy in a subinterval from a Gegenbauer partial sum of a piecewise analytic function, Math. Comp. 64:211 (1995), 10811095. CMP 95:11
 8.
 I. Gradshteyn and I. Ryzhik, Tables of Integrals, Series, and Products, Academic Press, 1980. MR 81g:33001
 9.
 F. John, Partial Differential Equations, SpringerVerlag, 1982. MR 87g:35002
 10.
 S. Orszag, Fourier Series on Spheres, Mon. Wea. Rev. 102 (1978), 5675.
 11.
 P. N. Swarztrauber, On the Spectral Approximation of Discrete Scalar and Vector Functions on a Sphere, Siam J. Numer. Anal., 16:6 (1979), 934949. MR 81c:65011
 12.
 H. Weyl, Die Gibbssche Erscheinung in der Theorie der Kugelfunktionen, , SpringerVerlag, 1968, 305320.
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/S0025571897008284
PII:
S 00255718(97)008284
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 F496209510074, NSF grant DMS 95008 14, and DARPA/ONR AASERT grant N000149310985.
Article copyright:
© Copyright 1997 American Mathematical Society
