Estimations of the remainder of spherical harmonic series

2008 ◽  
Vol 145 (1) ◽  
pp. 243-255 ◽  
Author(s):  
XIRONG CHANG ◽  
FENG DAI ◽  
KUNYANG WANG
Keyword(s):  
Spherical Harmonic ◽  
Remainder Term ◽  
Asymptotic Estimate ◽  
Harmonic Series ◽  
Main Term ◽  
Partial Sums ◽  
Order Of Growth ◽  
Approximation Of Functions ◽  
Spherical Harmonic Series ◽  
Precise Constant

AbstractAn asymptotic estimate is obtained for the error in approximation of functions by partial sums of spherical harmonic expansions. The precise constant in the main term is found and the order of growth of the remainder term is given.

On the computation and approximation of ultra-high-degree spherical harmonic series

2006 ◽  
Vol 81 (9) ◽  
pp. 603-615 ◽  
Author(s):  
Christopher Jekeli ◽  
Jong Ki Lee ◽  
Jay H. Kwon
Keyword(s):  
Spherical Harmonic ◽  
Harmonic Series ◽  
Spherical Harmonic Series ◽  
High Degree

scholarly journals Explicit Construction of the Modified Spherical Harmonic Series for the Gravity Gradients and Analysis of their Characteristics

2007 ◽  
Vol 42 (4) ◽  
pp. 185-214
Author(s):  
M. Petrovskaya ◽  
A. Vershkov
Keyword(s):  
Spherical Harmonic ◽  
Orthogonal Series ◽  
Explicit Construction ◽  
Anomalous Behavior ◽  
Harmonic Series ◽  
Geopotential Model ◽  
Orthogonal Expansion ◽  
Gravity Gradients ◽  
Spherical Harmonic Series ◽  
Degree Variances

Explicit Construction of the Modified Spherical Harmonic Series for the Gravity Gradients and Analysis of their CharacteristicsThe present paper complements the research carried out in PV2008 (Petrovskaya and Vershkov, 2008), concerning the expansion of the gravity gradients in the local north-oriented reference frame in orthogonal series of modified spherical harmonics. In PV2008 procedures are developed for recovering the orthogonal bases of these series. Then an idea is briefly described how the spectral relations can be obtained between the gravity gradients and the geopotential. However no explicit procedures are demonstrated for their derivation. In the present paper successive transformations are described for each derivative which convert the initial non-orthogonal expansion into the orthogonal series. The resulting spectral relations are applied for evaluating the harmonic coefficients of these series at different altitudes, on the basis of the geopotential model EGM2008. The corresponding degree variances are estimated. The new simple expressions for the gravity gradients are convenient for various applications. In the present paper they are implemented for constructing digital colored maps for Fennoscandia region which attracts much attention of geophysicists. These maps visually demonstrate an anomalous behavior of the gravity gradients in this area.

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