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. Geometric Tomography 1995 Groemer, H. , Fourier series and spherical harmonics in convexity, Handbook of

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harmonics in convexity, Handbook of convex geometry , ed. by P. M. Gruber and J. M. Wills, North-Holland, Amsterdam, 1993, 1259-1295. MR 94j :52001 Fourier series and spherical harmonics in convexity

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The topographic and atmospheric masses influence the satellite gravity gradiometry data, and it is necessary to remove these effects as precise as possible to make the computational space harmonic and simplify the downward continuation of such data. The topographic effects have been formulated based on constant density assumption for the topographic masses. However in this paper we formulate and study the effect of lateral density variation of crustal and topographic masses on the satellite gravity gradiometry data. Numerical studies over Fennoscandia and Iran show that the lateral density variation effect of the crust on GOCE data can reach to 1.5 E in Fennoscandia and 1 E in Iran. The maximum effect of lateral density variation of topography is 0.1 E and 0.05 E in Iran and Fennoscandia, respectively.

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Summary Classical representations of the Legendre polynomials in terms of generating functions are nonconvergent in the usual sense for extreme values of the generating parameter. A generalized type of convergence is explored and, in conjunction with the theory of Fourier series and spherical harmonics, it is applied to the computation of the fundamental solution of the Laplace-Beltrami operator on the n-sphere.

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Groemer, H. , Geometric Applications of Fourier Series and Spherical Harmonics , Encyclopedia of Mathematics and its Applications, vol. 61. Cambridge University Press, Cambridge 1996. MR 97j :52001

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Computational time is an important matter in numerical aspects and it depends on the algorithm and computer that is used. An inappropriate algorithm can increase computation time and cost. The main goal of this paper is to present a vectorization algorithm to speed up the global gradiometric synthesis and analysis. The paper discusses details of this technique and its very high capabilities. Numerical computations show that the global gradiometric synthesis with 0.5° × 0.5° resolution can be done in a few minutes (6 minutes) by vectorization, which is considerable less compared to several hours (9 hours) by an inappropriate algorithm. The global gradiometric analysis of representation by spherical harmonics up to degree and order of 360, can be performed within one hour using vectorization, but if an inconvenient algorithm is used it can be delayed more than 1 day. Here we present the vectorization technique to gradiometric synthesis and analysis, but it can also be used in many other computational aspects and disciplines.

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Essential Wavelets for Statistical Applications and Data Analysis Müller C 1966: Spherical Harmonics. Lecture Notes in Mathematics, 17, Springer, Berlin, Heidelberg

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Rummel R, Sanso F, Gelderen M, Koop R, Schrama E, Brovelli M, Migiliaccio F, Sacerdote F 1993: Spherical harmonic analysis of satellite gradiometry. Publ. Geodesy, New Series, No. 39, Netherlands Geodetic Commission, Delft

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harmonics , Lecture Notes in Mathematics, 17 , Springer-Verlag, Berlin-New York, 1966. MR 33 #7593 Spherical harmonics Nievergelt, Y

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