Performing exactly the same calculation on the different GRACE spherical harmonics
leads to slightly different results.
it is possible to express the ODF f in a (generally infinite) sum of 'spherical harmonics
', where the spherical harmonics
are the two-dimensional analogues to the sines and cosines in the Fourier series (see Section 2).
We compared the produced TWSA time series with three standard monthly gridded GRACE TWSA products from CSR, GFZ, and JPL, all based on RL05 spherical harmonics
To alleviate this problem, a spherical harmonics
expansion-based MLFMA (SE-MLFMA) for the dyadic form of surface integral equation (SIE) was proposed .
(15) cannot be resolved by traditional expansion in spherical harmonics
with coefficients given by the monomials [r.sup.l].
In this work [S.sub.lm] ([theta], [phi]) is the complex ([S.sub.lm] = [Y.sub.lm]) or real spherical harmonics
. The definition of phases for the complex spherical harmonics
([Y.sup.*.sub.lm] = [Y.sub.l-m]) differs from the Condon- Shortley phases  by the sign factor [(-1).sup.m] and can be defined as
On the one hand, there is a conceptually very appealing view on spherical harmonics
, due to Lebeau and Guillemin, based on the Szegoo kernel of the Fermat quadric.
It is easy to show that the [sup.1][Y.sup.ls]-RNSSTH and [sup.2][Y.sup.ls]- MRNSSTH harmonics for s = 0 are transformed into the scalar spherical harmonics
Based on the LO data, the spherical harmonics
of the Moon (degree 4 in the tesserals and degree 8 in the zonals) were first published in 1968 ; later that year, the existence of large mascons under the lunar ringed maria was also discovered .
Garboczi (2002) was probably the first employing the spherical harmonics
(SH) technique to mathematically describe in three-dimensions particle shapes applying it to aggregates used in concrete.
The ellipsoidal harmonics in these spheroconical coordinates replace the spherical harmonics
of spherical polar coordinates.
Among their topics are summaries of finite difference method and pseudospectral methods, global Radial basis functions for solving partial differential equations, applications to geo-modeling in spherical domains, spherical harmonics
, and some node distribution strategies.