# Geometric Applications Of Fourier Series And Spherical Harmonics Pdf

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- Geometric Applications of Fourier Series and Spherical Harmonics
- Geometric applications of Fourier series and spherical harmonics
- Geometric applications of Fourier series and spherical harmonics

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## Geometric Applications of Fourier Series and Spherical Harmonics

Computations of Fourier coefficients and related integrals of the associated Legendre functions with a new method along with their application to spherical harmonics analysis and synthesis are presented.

The method incorporates a stable three-step recursion equation that can be processed separately for each colatitudinal Fourier wavenumber. Recursion equations for the zonal and sectorial modes are derived in explicit single-term formulas to provide accurate initial condition. Stable computations of the Fourier coefficients as well as the integrals needed for the projection of Legendre functions are demonstrated for the ultra-high degree of 10, corresponding to the resolution of one arcmin.

Fourier coefficients, computed in double precision, are found to be accurate to 15 significant digits, indicating that the normalized error is close to the machine round-off error. The Legendre function of degree 10, and order 5,, synthesized from Fourier coefficients, is accurate to the machine round-off error. Further extension of the method to even higher degrees seems to be realizable without significant deterioration of accuracy.

The Fourier series is applied to the projection of Legendre functions to the high-resolution global relief data of the National Geophysical Data Center of the National Oceanic and Atmospheric Administration, and the spherical harmonic degree variance power spectrum of global relief data is discussed.

This is a preview of subscription content, access via your institution. Rent this article via DeepDyve. Blais JAR Discrete spherical harmonic transforms: numerical preconditioning and optimization. LNCS, vol Springer, Berlin, pp — Cheong HB Double Fourier series on a sphere: applications to elliptic and vorticity equations.

J Comput Phys — Stud Geophys Geod — Google Scholar. Holmes SA, Featherstone WE A unified approach to the Clenshaw summation and the recursive computation of very high degree and order normalised associated Legendre functions. J Geod — Iwanami Shoten, Tokyo, p Nehrkorn T On the computation of Legendre functions in spectral models.

Mon Weather Rev — Risbo T Fourier transform summation of Legendre series and D-functions. Swarztrauber PN The vector harmonic transform method for solving partial differential equations in spherical geometry.

Wittwer T, Klees R, Seitz K Ultra-high degree spherical harmonic analysis and synthesis using extended range arithmetic. Download references. Correspondence to Hyeong-Bin Cheong.

Reprints and Permissions. Cheong, HB. Fourier-series representation and projection of spherical harmonic functions. J Geod 86, — Download citation. Received : 21 June Accepted : 20 March Published : 11 April Issue Date : November Search SpringerLink Search. Abstract Computations of Fourier coefficients and related integrals of the associated Legendre functions with a new method along with their application to spherical harmonics analysis and synthesis are presented. Immediate online access to all issues from Subscription will auto renew annually.

References Blais JAR Discrete spherical harmonic transforms: numerical preconditioning and optimization. Springer, Berlin, pp — Cheong HB Double Fourier series on a sphere: applications to elliptic and vorticity equations. View author publications. Rights and permissions Reprints and Permissions. About this article Cite this article Cheong, HB.

## Geometric applications of Fourier series and spherical harmonics

In mathematics and physical science , spherical harmonics are special functions defined on the surface of a sphere. They are often employed in solving partial differential equations in many scientific fields. Since the spherical harmonics form a complete set of orthogonal functions and thus an orthonormal basis , each function defined on the surface of a sphere can be written as a sum of these spherical harmonics. This is similar to periodic functions defined on a circle that can be expressed as a sum of circular functions sines and cosines via Fourier series. Like the sines and cosines in Fourier series, the spherical harmonics may be organized by spatial angular frequency , as seen in the rows of functions in the illustration on the right.

With appropriate weights, one cycle or period of the summation can be made to approximate an arbitrary function in that interval or the entire function if it too is periodic. As such, the summation is a synthesis of another function. The discrete-time Fourier transform is an example of Fourier series. The process of deriving weights that describe a given function is a form of Fourier analysis. For functions on unbounded intervals, the analysis and synthesis analogies are Fourier transform and inverse transform.

Computations of Fourier coefficients and related integrals of the associated Legendre functions with a new method along with their application to spherical harmonics analysis and synthesis are presented. The method incorporates a stable three-step recursion equation that can be processed separately for each colatitudinal Fourier wavenumber. Recursion equations for the zonal and sectorial modes are derived in explicit single-term formulas to provide accurate initial condition. Stable computations of the Fourier coefficients as well as the integrals needed for the projection of Legendre functions are demonstrated for the ultra-high degree of 10, corresponding to the resolution of one arcmin. Fourier coefficients, computed in double precision, are found to be accurate to 15 significant digits, indicating that the normalized error is close to the machine round-off error.

GEOMETRIC APPLICATIONS OF FOURIER SERIES AND SPHERICAL HARMONICS (Encyclopedia of Mathematics and its Applications 61).

## Geometric applications of Fourier series and spherical harmonics

Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. It only takes a minute to sign up. I need a topic, a primer, that will be able to introduce me to spherical harmonics and how to translate and use them with the usual tools of linear algebra and calculus, namely matrices, polynomials and derivatives for example. What topics do you suggest I should touch to get up and running with spherical harmonics starting with a linear algebra and calculus background?

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*This is the first comprehensive exposition of the application of spherical harmonics to prove geometric results. The author presents all the necessary tools from classical theory of spherical harmonics with full proofs. Groemer uses these tools to prove geometric inequalities, uniqueness results for projections and intersection by planes or halfspaces, stability results, and characterizations of convex bodies of a particular type, such as rotors in convex polytopes.*

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