# Geometric Applications Of Fourier Series And Spherical Harmonics Pdf

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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.

## Fourier-series representation and projection of spherical harmonic functions

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? I would recommend studying some basics on Fourier analysis, lie algebras and Laplace-Beltrami operators.

## 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.

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Enter your mobile number or email address below and we'll send you a link to download the free Kindle App. Then you can start reading Kindle books on your smartphone, tablet, or computer - no Kindle device required. To get the free app, enter your mobile phone number. 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 half-spaces, stability results, and characterizations of convex bodies of a particular type, such as rotors in convex polytopes.

## Geometric applications of Fourier series and spherical harmonics

Сьюзан представила себе Хейла в западне, в окутанной паром ловушке. Может быть, он что-нибудь поджег. Она посмотрела на вентиляционный люк и принюхалась. Но запах шел не оттуда, его источник находился где-то поблизости. Сьюзан посмотрела на решетчатую дверь, ведущую в кухню, и в тот же миг поняла, что означает этот запах.

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Хейл очень опасен. Он… Но Стратмор растворился в темноте. Сьюзан поспешила за ним, пытаясь увидеть его силуэт. Коммандер обогнул ТРАНСТЕКСТ и, приблизившись к люку, заглянул в бурлящую, окутанную паром бездну. Молча обернулся, бросил взгляд на погруженную во тьму шифровалку и, нагнувшись приподнял тяжелую крышку люка. Она описала дугу и, когда он отпустил руку, с грохотом закрыла люк.

Может быть, все-таки скажете что-нибудь. Что помогло бы мне? - сказал Беккер. Росио покачала головой: - Это. Но вам ее не найти.

Понятия не имею. - Похож на китайца. Японец, подумал Беккер.

Fantina M.

This is the first comprehensive exposition of the application of spherical harmonics to prove geometric results.

Kimmy2345