We perform a complete study of the truncation error of the Jacobi-Anger series. This series expands every plane wave in terms of spherical harmonics . We consider the truncated series where the summation is performed over the ’s satisfying . We prove that if is large enough, the truncated series gives rise to an error lower than as soon as satisfies where is the Lambert function and are pure positive constants. Numerical experiments show that this asymptotic is optimal. Those results are useful to provide sharp estimates for the error in the fast multipole method for scattering computation.
@article{M2AN_2004__38_2_371_0, author = {Carayol, Quentin and Collino, Francis}, title = {Error estimates in the fast multipole method for scattering problems. Part 1 : truncation of the Jacobi-Anger series}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique}, volume = {38}, year = {2004}, pages = {371-394}, doi = {10.1051/m2an:2004017}, mrnumber = {2069152}, zbl = {1077.41027}, language = {en}, url = {http://dml.mathdoc.fr/item/M2AN_2004__38_2_371_0} }
Carayol, Quentin; Collino, Francis. Error estimates in the fast multipole method for scattering problems. Part 1 : truncation of the Jacobi-Anger series. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 38 (2004) pp. 371-394. doi : 10.1051/m2an:2004017. http://gdmltest.u-ga.fr/item/M2AN_2004__38_2_371_0/
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