By Willi Freeden, Volker Michel, Birkhauser

This self-contained text/reference offers a simple starting place for practitioners, researchers, and scholars attracted to any of the various components of multiscale (geo)potential thought. New mathematical tools are developed enabling the gravitational capability of a planetary physique to be modeled utilizing a continuing stream of observations from land or satellite tv for pc units. Harmonic wavelets equipment are brought, in addition to speedy computational schemes and numerous numerical try examples. offered are multiscale techniques for varied geoscientific difficulties, together with geoidal decision, magnetic box reconstruction, deformation research, and density edition modelling

With workouts on the finish of every bankruptcy, the book may be used as a textbook for graduate-level classes in geomathematics, utilized arithmetic, and geophysics. The work is usually an updated reference textual content for geoscientists, utilized mathematicians, and engineers.

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M, k=1, ... 144) N = L alKHarmo, .. ,rn(fI)(~' 'fll), 1=1 where KHarmo, .. 145) given by ~,'fl E n, is the reproducing kernel function of the space Harmo, ... ,m(n). 2. 147) HE L1[-I, +1], Y n E Harmn(D), where the "Legendre transform" of HE L1[-I, +1] is given by HI\ L2 r- 1,+11 (n) = (H, Pn )V[-l,+l] = 27r [:1 H(t)Pn(t) dt. " The spherical harmonics Y n are the eigenfunctions of the integral operator corresponding to the eigenvalues HI\ L2 r- 1,+11 (n). Therefore, the Funk-Heeke formula simplifies most manipulations with spherical harmonics.

N, imply Y = 0. Then the set X N is called a Harmp, ... ,q(O)-fundamental system on O. Note that the property of X 2n +1 being a Harmn(O)-fundamental system implies that Y E Harmn(O) with Y('fli) = 0, i = 1, ... , 2n + 1, holds if and only if Y = 0. Assume that XN = {'fl1, ... , 'flN} is a Harmo, ... ,m (O)-fundamental system on n. Furthermore, suppose that Y is an element of class Harmo, ... 142) (a1, ... , aN) T, of the linear system N LaIYn,k('flI)=Cn,k, 1=1 n=O, ... ,m, k=1, ... 144) N = L alKHarmo, ..

1*. 97) where "1*. = div* and L*· = curl*, respectively, denote the surface divergence and the surface curl given by 3 "1~ . i(e) . 98) i=l and 3 L~ . i(e) . 2. 3. The aforementioned relations can be understood from the well-known role of the Beltrami operator D. * in the representation of the Laplace operator D. 101) In spherical coordinates the operators D. ~ = \7* - ~ * L~ = - ata (1 - ata + 1 -1 t2 (a) a'P 2' 'P 1 a ~a € JI=t2 a'P + € Y 1 - t- at' t 2) t ~a 1- t- at _€'Py t 1 a + € JI=t2 a'P .