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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...
We perform a complete study of the truncation error of the Gegenbauer series. This series yields an expansion of the Green kernel of the Helmholtz equation, , which is the core of the Fast Multipole Method for the integral equations. We consider the truncated series where the summation is performed over the indices . 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, depends only on and are...
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 ϵ
as soon as L satisfies
where W is the Lambert function and
are pure positive constants.
Numerical experiments show that this
asymptotic is...
We perform a complete study
of the truncation error of the Gegenbauer series.
This series yields an expansion of the Green kernel of the
Helmholtz equation,
,
which is the core of the Fast Multipole Method for the integral equations.
We consider the truncated series where the summation is
performed over the indices .
We prove that if is large enough,
the truncated series gives rise to an error lower than ϵ
as soon as L satisfies
where W is the Lambert function,
depends only on...
We obtain some approximate identities whose accuracy depends on the bottom of the discrete spectrum of the Laplace-Beltrami operator in the automorphic setting and on the symmetries of the corresponding Maass wave forms. From the geometric point of view, the underlying Riemann surfaces are classical modular curves and Shimura curves.
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