Re: LEGENDRE Fourier Series

From: Bruce Scott TOK (Use-Author-Supplied-Address-Header_at_[127.1)
Date: 08/16/04 

Date: Mon, 16 Aug 2004 13:19:48 +0200 (MEST)

One Usenet Poster wrote:

|> Faisal wrote:
|>
|> > Hello Friendz ,
|> >
|> > Well i have heard that A FUNCTION CAN BE REPRESENTED IN TERMS OF
|> > LEGENDRE POLYNOMIALS , WHICH FORM A COMLETE SET OF MUTUALLY ORTHOGONAL
|> > FUNCTIONS OVER INTERVAL -1 < t < +1
|> > Can any1 tell me the derivation of this ( whole derivation ) and a
|> > desription of it ....i means where it is applicable in real world n
|> > things related to it

Look up proofs on orthogonal polynomials...

(note to others: he is not looking for polynomial _approximations_)

Depending on boundary conditions, a function may be decomposed into a
series of functions which satisfy two properties: orthogonality and
completeness. The most familiar example is periodic slab geometry and
sines and cosines (Fourier series).

On [-1,1] with free boundaries the Legendre polynomials satisfy this
property, and for spherical geometry extending to infinity...

|> The Earth's gravitational potential is usually modeled as a spherical
|> harmonic expansion using Legendre polynomials. A quick Google search on
|> any of these terms will yield numerous sources.

..this combination of exponentials (for longitude) and Legendre
polynomials (for sine of latitude) do it. 

-- 
cu,
Bruce
drift wave turbulence:  http://www.rzg.mpg.de/~bds/
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