Representation of Grad's Hermite polynomials as sums of products of sonine polynomials and solid spherical harmonics

1962 ◽  
Vol 9 (1) ◽  
pp. 255-259 ◽  
Author(s):  
Ernest Ikenberry
Keyword(s):  
Spherical Harmonics ◽  
Hermite Polynomials ◽  
Solid Spherical Harmonics ◽  
Sums Of Products

scholarly journals The topographic bias in Stokes’ formula vs. the error of analytical continuation by an Earth Gravitational Model - are they the same?

2015 ◽  
Vol 5 (1) ◽  
Author(s):  
L. E. Sjöberg
Keyword(s):  
Gravity Anomaly ◽  
Spherical Harmonics ◽  
Analytical Continuation ◽  
Geoid Determination ◽  
Disturbing Potential ◽  
Solid Spherical Harmonics ◽  
Topographic Bias ◽  
Gravitational Model ◽  
Stokes Formula ◽  
Original Formula

AbstractGeoid determination below the topographic surface in continental areas using analytical continuation of gravity anomaly and/or an external type of solid spherical harmonics determined by an Earth GravitationalModel (EGM) inevitably leads to a topographic bias, as the true disturbing potential at the geoid is not harmonic in contrast to its estimates. We show that this bias differs for the geoid heights represented by Stokes’ formula, an EGMand for the modified Stokes formula. The differences are due to the fact that the EGM suffers from truncation and divergence errors in addition to the topographic bias in Stokes’ original formula.

Expansion theorems for solid spherical harmonics

1965 ◽  
Vol 9 (2) ◽  
pp. 175-178 ◽  
Author(s):  
J.P. Dahl ◽  
M.P. Barnett
Keyword(s):  
Spherical Harmonics ◽  
Solid Spherical Harmonics

Translation of real solid spherical harmonics

2012 ◽  
Vol 113 (10) ◽  
pp. 1544-1548 ◽  
Author(s):  
Jaime Fernández Rico ◽  
Rafael López ◽  
Ignacio Ema ◽  
Guillermo Ramírez
Keyword(s):  
Spherical Harmonics ◽  
Solid Spherical Harmonics ◽  
Real Solid

A Constructive Proof of Helmholtz’s Theorem

2019 ◽  
Vol 72 (4) ◽  
pp. 521-533
Author(s):  
Bernardo De La Calle Ysern ◽  
José C Sabina De Lis
Keyword(s):  
Vector Field ◽  
Spherical Harmonics ◽  
Constructive Proof ◽  
Newtonian Potential ◽  
Hölder Continuous ◽  
Open Set ◽  
Solid Spherical Harmonics ◽  
Local Solutions

Summary It is a known result that any vector field ${\boldsymbol{u}}$ that is locally Hölder continuous on an arbitrary open set $\Omega\subset \mathbb{R}^3$ can be written on $\Omega$ as the sum of a gradient and a curl. Should $\Omega$ be unbounded, no conditions are required on the behaviour of ${\boldsymbol{u}}$ at infinity. We present a direct, self-contained proof of this theorem that only uses elementary techniques and has a constructive character. It consists in patching together local solutions given by the Newtonian potential that are then modified by harmonic approximations—based on solid spherical harmonics—to assure convergence near infinity for the resulting series.

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