Filtering on the unit sphere using spherical harmonics

This paper tackles the construction of fractal maps on the unit sphere. The functions defined are a generalization of the classical spherical harmonics. The methodology used involves an iterated function system and a linear and bounded operator of functions on the sphere. For a suitable choice of the coefficients of the system, one obtains classical maps on the sphere. The different values of the system parameters provide Bessel sequences, frames, and Riesz fractal bases for the Lebesgue space of the square integrable functions on the sphere. The Laplace series expansion is generalized to a sum in terms of the new fractal mappings.

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Density Function of the FK4 Systematic Errors

Symposium - International Astronomical Union ◽

10.1017/s0074180900086368 ◽

1990 ◽

Vol 141 ◽

pp. 94-94

Author(s):

V. S. Gubanov ◽

N. I. Solina

Keyword(s):

Density Function ◽

Spherical Harmonics ◽

Unit Sphere ◽

Systematic Errors ◽

Layer Potential ◽

Partial Derivatives ◽

Derivatives Of

A notion of density function of systematic errors of astrometric catalogues distributed on the unit sphere as a simple layer is introduced. Components of the catalogue errors in any direction are determined as partial derivatives of layer potential in the same direction. For example, the density function of FK4 errors is computed as an expansion of spherical harmonics.

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Weighted Approximation for Jackson-Matsuoka Polynomials on the Sphere

Abstract and Applied Analysis ◽

10.1155/2012/302853 ◽

2012 ◽

Vol 2012 ◽

pp. 1-11

Author(s):

Guo Feng

Keyword(s):

Spherical Harmonics ◽

Unit Sphere ◽

Best Approximation ◽

Weighted Approximation ◽

Modulus Of Smoothness ◽

The Best Approximation

We consider the best approximation by Jackson-Matsuoka polynomials in the weightedLpspace on the unit sphere ofRd. Using the relation betweenK-functionals and modulus of smoothness on the sphere, we obtain the direct and inverse estimate of approximation by these polynomials for theh-spherical harmonics.

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Orthogonal Polynomials for a Family of Product Weight Functions on the Spheres

Canadian Journal of Mathematics ◽

10.4153/cjm-1997-009-4 ◽

1997 ◽

Vol 49 (1) ◽

pp. 175-192 ◽

Cited By ~ 29

Author(s):

Yuan Xu

Keyword(s):

Orthogonal Polynomials ◽

Spherical Harmonics ◽

Unit Sphere ◽

Poisson Kernel ◽

Reflection Group ◽

Weight Functions ◽

Intertwining Operator ◽

Finite Reflection Group ◽

Explicit Formulae ◽

Product Weight

AbstractBased on the theory of spherical harmonics for measures invariant under a finite reflection group developed by Dunkl recently, we study orthogonal polynomials with respect to the weight functions |x1|α1 . . . |xd|αd on the unit sphere Sd-1 in ℝd. The results include explicit formulae for orthonormal polynomials, reproducing and Poisson kernel, as well as intertwining operator.

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Spherical Harmonics and Approximations on the Unit Sphere: An Introduction

10.1007/978-3-642-25983-8 ◽

2012 ◽

Cited By ~ 91

Author(s):

Kendall Atkinson ◽

Weimin Han

Keyword(s):

Spherical Harmonics ◽

Unit Sphere

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Zonal and tesseral harmonic perturbations of an artificial satellite

Symposium - International Astronomical Union ◽

10.1017/s0074180900105625 ◽

1966 ◽

Vol 25 ◽

pp. 323-325 ◽

Cited By ~ 1

Author(s):

B. Garfinkel

Keyword(s):

Angular Velocity ◽

Spherical Harmonics ◽

Artificial Satellite ◽

Compact Form ◽

Earth’S Rotation ◽

Earth's Rotation ◽

Secular Variations ◽

Tesseral Harmonics

The paper extends the known solution of the Main Problem to include the effects of the higher spherical harmonics of the geopotential. The von Zeipel method is used to calculate the secular variations of orderJmand the long-periodic variations of ordersJm/J2andnJm,λ/ω. HereJmandJm,λare the coefficients of the zonal and the tesseral harmonics respectively, withJm,0=Jm, andωis the angular velocity of the Earth's rotation. With the aid of the theory of spherical harmonics the results are expressed in a most compact form.

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