Proof that the legendre polynomials are orthogonal polynomials

In summary, Legendre polynomials are a set of orthogonal polynomials that have special properties when used in numerical integration. They are orthogonal to all other polynomials of lower degree with respect to a weight function, and can be used as a basis for all polynomials of degree less than or equal to n. This is proven by showing that any polynomial of degree m can be represented as a linear combination of Legendre polynomials of degree at most m, and thus the only surviving term in the inner product will be the nth one.
  • #1
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I'm now studying the application of legendre polynomials to numerical integration in the so called gaussian quadrature. There one exploits the fact that an orthogonal polynomial of degree n is orthogonal to all other polynomials of degree less than n with respect to some weight function. For legendre polynomials that must mean that

[tex]\int_{-1}^{1} L_n(x) P_m(x) dx = 0[/tex]

for all P(x) where m is less than n. How does one prove that the legendre polynomials are in the set of such orthogonal polynomials? It's okay that they are orthogonal among themselves, but I wonder how to show that they are orthogonal to everyone else with lower degree?
 
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  • #2
How does one prove that the legendre polynomials are in the set of such orthogonal polynomials?

This is unclear. What is the set you are referring to?

It's okay that they are orthogonal among themselves, but I wonder how to show that they are orthogonal to everyone else with lower degree?

Any polynomial of degree m can be represented as a linear combination of Legendre polynomials of degree at most m.
 
  • #3
show that the legendre polynomials of degree ≤ n, are linearly independent, and thus form a basis for all polynomials of degree ≤ n.

therefore, every polynomial of degree ≤ n can be written as a linear combination of the Lj (j = 0,1,2,...,n):

[tex]P_m(x) = a_0L_0(x) + \dots + a_nL_n(x)[/tex]

which will make the only surviving term in the inner product the nth one:

[tex]\int_{-1}^1L_n(x)a_nL_n(x) dx[/tex]

but if Pm is of degree < n, the coefficent an of Ln will be 0.
 
  • #4
Ah, thanks! That was the argument I was looking for.
 

Related to Proof that the legendre polynomials are orthogonal polynomials

1. What are Legendre polynomials?

Legendre polynomials are a set of orthogonal polynomials named after French mathematician Adrien-Marie Legendre. They are used in various mathematical and scientific applications, such as approximating complex functions and solving differential equations.

2. Why are Legendre polynomials important?

Legendre polynomials have many important properties, such as orthogonality, which make them useful in solving various mathematical problems. They also have applications in physics, engineering, and other fields.

3. How do you prove that Legendre polynomials are orthogonal?

The orthogonality of Legendre polynomials can be proven using various methods, such as the Gram-Schmidt process, which involves constructing a set of orthogonal polynomials from a given set of polynomials. Another method is by using the inner product of two polynomials and showing that it equals zero if the two polynomials are different.

4. Can Legendre polynomials be used to approximate any function?

Yes, Legendre polynomials can be used to approximate any function. This is because they form a complete orthogonal system, which means that any function can be expressed as a linear combination of Legendre polynomials.

5. What are some applications of Legendre polynomials?

Legendre polynomials have applications in various fields, such as physics, engineering, and statistics. They are used in solving differential equations, approximating complex functions, and in numerical analysis methods like the finite element method. They are also used in signal processing, image processing, and data analysis.

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