# Legendre Polynomials as an Orthogonal Basis

• physconomic
In summary: So ##P_2## is a polynomial of degree (n=2) that is orthogonal to all polynomials of degree <n (i.e. degree strictly less than 2).And so forth. Note that the Legendre polynomials up to degree n form a basis for the space of all polynomials up to degree n. And that's how you show that the inner product between ##P_n## and any polynomial of degree <n is 0. And that's how you find the polynomials ##P_0...P_4##: Just apply the Gram-Schmidt orthogonalization procedure to the power basis up to degree 4.In summary, the conversation discusses the properties of Legend
physconomic
Homework Statement
If you had legendre polynomials defined in ##L^2([-1,1])##, with ##||Pn_2||^2=\frac{2}{2n+1}##, show that for any polynomial with p a set of ##L^2([-1,1])##, with degree less than n, we have the inner product of ##P_n## and p = 0. Find the polynomials ##P_0,... P_4##
Relevant Equations
Integral form of inner product
If you had legendre polynomials defined in ##L^2([-1,1])##, with ##||Pn_2||^2=\frac{2}{2n+1}##, show that for any polynomial with p a set of ##L^2([-1,1])##, with degree less than n, we have the inner product of ##P_n## and p = 0. Find the polynomials ##P_0,... P_4##

Tried to use the integral form but getting no where with it

Your (source's) wording is confusing. Specifically: "show that for any polynomial with p a set of... less than ##n##". What is p here?
Does it mean: "with p a set of polynomials in ##L^2([-1,1])## with degree less than n"?
Does it mean" "with p in a set of Legendre polynomials in ..."?

Legendre polynomials are mutually orthogonal under the ##L^2([-1,1])## norm and indeed you can derive them fairly easily up to scalar multipliers by the Gram-Schmidt procedure applied to the power basis ##\{ 1,x,x^2,x^3, \ldots\}##.

Starting with ##P_0(x)=1## and ##P_1(x)=x## (already orthogonal), then ##p_2(x)=x^2## an
## \langle p_2,P_1\rangle = 0## but:
$$\langle p_2,P_0\rangle = \int_{-1}^1 1\cdot x^2 dx = 2/3$$
So we take:
## P_2 = p_2 -\frac{\langle P_0,p_2\rangle}{\langle P_0,P_0\rangle}\cdot P_0##
and that linear combination will be orthogonal to ##P_0##.
In this case with ##\langle P_0,P_0\rangle = 2## we get:
$$P_2 = p_2 - \frac{1}{3}P_0,\quad P_2(x) = x^2 -\frac{1}{3}$$
However with standard normalization the actual degree 2 Legrange polynomial is 3/2 times this one, namely ##P_2(x)=\frac{1}{2}(3x^2-1)##. (Standard normalization assures ##P_n(1)=1, P_n(-1)=(-1)^n##.)

That's all I can say without better context to understand the ?mis-worded? question.

physconomic
Immediate followup. Here's a point to remember which may help understand the question and its solution:
The Legendre polynomials up to degree n form a basis for the space of all polynomials up to degree n.

Note in my prior post that while ##p_2: p_2(x)=x^2## was not orthogonal to ##P_0##, the constructed ##P_2## was orthogonal to both ##P_0## and ##P_1## (and thus to any linear combination of them >wink wink<).

physconomic

## 1. What are Legendre polynomials?

Legendre polynomials are a set of orthogonal polynomials that are used as a basis for representing functions in mathematical analysis. They were first introduced by French mathematician Adrien-Marie Legendre in the late 18th century.

## 2. What does it mean for a basis to be orthogonal?

A basis is considered orthogonal when the inner product of any two basis vectors is equal to zero, meaning they are perpendicular to each other. This property is useful in mathematical analysis as it allows for simpler calculations and easier representation of functions.

## 3. How are Legendre polynomials used as a basis?

Legendre polynomials are used as a basis for representing functions in a process called the Legendre expansion. This involves expressing a given function as a linear combination of Legendre polynomials, with each polynomial multiplied by a coefficient. This allows for the function to be approximated and analyzed in a simpler form.

## 4. What are the advantages of using Legendre polynomials as a basis?

One of the main advantages of using Legendre polynomials as a basis is their orthogonality, which simplifies calculations and allows for efficient representation of functions. They are also widely applicable, as they can be used to represent a wide range of functions and have been studied extensively in mathematics.

## 5. Are there any limitations to using Legendre polynomials as a basis?

While Legendre polynomials have many advantages, they also have some limitations. They are only orthogonal on a specific interval, typically from -1 to 1, and may not be suitable for representing functions outside of this range. Additionally, the Legendre expansion may not always converge to the exact function, but rather provides an approximation.

• Calculus
Replies
2
Views
1K
• Calculus and Beyond Homework Help
Replies
5
Views
2K
Replies
1
Views
991
Replies
1
Views
1K
• Differential Equations
Replies
1
Views
2K
• Differential Equations
Replies
1
Views
1K
• Topology and Analysis
Replies
2
Views
2K
• Calculus and Beyond Homework Help
Replies
1
Views
1K
• Calculus
Replies
15
Views
3K
• Calculus and Beyond Homework Help
Replies
2
Views
418