Talk:Legendre polynomials

From Wikipedia, the free encyclopedia

[edit] Applications of the Legendre polynomials

Can anyone provide more concrete examples of applications of the Legendre polynomials, or when the Legendre differential equation arises? 171.64.133.56 22:48, 24 February 2006 (UTC)

In fact, they form a basis for the space of piecewise continuous functions defined on this interval, so any such function can be written as a linear combination of Legendre polynomials.

These can't be bases and linear combinations in the sense of linear algebra. Are we talking Hilbert space basis here?

Also, is it correct that P_n solves the differential equation with parameter n, or is n used in two different senses here? AxelBoldt 23:22 Feb 12, 2003 (UTC)

The Legendre polynomials form an orthogonal basis for the Hilbert space of square-integrable functions on the interval from -1 to 1, so we're talking about a "linear combination" in the Hilbert-space sense, i.e., not a finite linear combination. So it's easy to see that if sn is the nth partial sum, then

\lim_{n\rightarrow\infty}\int_{-1}^1\left|s_n(x)-f(x)\right|^2\,dx=0.

But if you want pointwise convergence to a piecewise continuous function, that takes more work. The person who wrote those words doesn't seem aware of the difference kinds of convergence or of the fact that mathematicians may often construe "linear combination" to mean finite linear combination. Michael Hardy 22:11 Mar 13, 2003 (UTC)

Yes, well, Hilbert spaces should be discussed in the article on orthogonal polynomials anyway, so as to cover the whole kit-n-kaboodle. linas 05:32, 28 Mar 2005 (UTC)

Usually the singular makes more sense that the plural in the title of an article, but in this case, the title "Legendre polynomial" makes the same amount of sense as "Beattle" as the title of an article about John, Paul, George, and Ringo. The whole sequence Legendre polynomials is to be thought of as a unit. Michael Hardy 22:15 Mar 13, 2003 (UTC)

Despite the style pronouncements at [1], Legendre polynomial is not appropriate as a title for an article about the

Legendre polynomials

(which see!). An article about John, Paul, George, and Ringo would not be titled Beatle, but Beatles' or The Beatles. So it is here. One does speak of a "Legendre polynomial" in the singular in some contexts, but generally those are of interest only because this polynomial sequence, like others, is thought of as a unit.

[edit] Associated Legendre functions?

My quantum mechanics text says speaks of something called the associated Legendre functions, which appears to be distinct from the associated Legendre polynomials. They have the form:

f_{lm}(\theta) = \frac{(\sin\theta)^{|m|}}{2^l l!} \left [ \frac{d}{d(\cos\theta)} \right ]^{l+|m|}(\cos^2(\theta)-1)^l

Is this something that somehow missed having an article? --Smack (talk) 05:46, 7 May 2005 (UTC)

No, you just didn't read the article carefully enough. Associated Legendre polynomials clearly mentions x = cosθ. linas 16:55, 8 May 2005 (UTC)

[edit] Standardization

I have reverted the word "normalize" back to "standardize", since the polynomials are not normal, and the standardization is just a convention. I'm still not happy with this -- it's a conflict between saying exactly the right thing vs. belaboring a point that isn't really important. Improvements? William Ackerman 23:38, 14 March 2006 (UTC)

Works for me. Nothing wrong with adding a sentance that says: "Sometimes this is called a "normalization", although the correct meaning of normalization is that the integral of the square of the function should be unity." linas 00:14, 15 March 2006 (UTC)
I still think normalized is the correct word here. Normalization does not necessarily mean to transform something to norm one, it is used for any transformation to get something in standard form. However, it is not that important. I do wonder though why the word "standardize" is written in bold. -- Jitse Niesen (talk) 09:16, 15 March 2006 (UTC)