Question

Express each of the following polynomials as linear combinations of Legendre polynomials. Hint: Start with the highest power of \(x\) and work down in finding the correct combination. \(3 x^{2}+x-1\)

Short answer

The polynomial can be written as \( 3x^2 + x - 1 = 2P_2(x) + P_1(x) - P_0(x) \).

Step-by-step solution

Identify Legendre Polynomials

Legendre polynomials are a sequence of orthogonal polynomials. The first few Legendre polynomials are:\[ P_0(x) = 1 \]\[ P_1(x) = x \]\[ P_2(x) = \frac{1}{2}(3x^2 - 1) \]

Rewrite the Polynomial in Terms of Legendre Polynomials

The given polynomial is \( 3x^2 + x - 1 \). We need to express this as a linear combination of the Legendre polynomials \( P_0(x), P_1(x), \) and \( P_2(x) \). Start with the highest degree term, which is \( 3x^2 \).

Match the Highest Degree Term

We know that \( P_2(x) = \frac{1}{2}(3x^2 - 1) \). Therefore, \( 3x^2 = 2P_2(x) + 1 \). Notice that the term \( 1 \) can be adjusted when we sum up the Legendre polynomials later.

Combine to Match All Terms

The polynomial can thus be written as:\[ 3x^2 + x - 1 = 2P_2(x) + 1 + P_1(x) - P_0(x) \]

Simplify

Simplify to get:\[ 3x^2 + x - 1 = 2P_2(x) + P_1(x) - P_0(x) \]

Key concepts

Orthogonal Polynomials

Orthogonal polynomials are a special class of polynomials that are orthogonal to each other with respect to a given inner product. This means that the integral of the product of two different polynomials from this sequence over a certain interval is zero. Legendre polynomials are a classic example of orthogonal polynomials. They are particularly useful in solving problems involving series expansions and polynomial transformations.
Legendre polynomials, denoted as \(P_n(x)\), are defined on the interval \([-1, 1]\). For example:

• \(P_0(x) = 1\)
• \(P_1(x) = x\)
• \(P_2(x) = \frac{1}{2}(3x^2 - 1)\)

Understanding the properties of orthogonal polynomials can help in simplifying complex polynomials and solving differential equations.

Linear Combination

A linear combination involves adding together scaled versions of some functions, polynomials, or vectors. In this context, expressing one polynomial as a linear combination of Legendre polynomials means finding coefficients such that:
\(a_0P_0(x) + a_1P_1(x) + a_2P_2(x) + ... = 3x^2 + x - 1\)
From the solution, we know:
\(3x^2 = 2P_2(x) + P_1(x) + P_0(x)\)
Each coefficient \(a_n\) corresponds to a Legendre polynomial \(P_n(x)\), making it possible to reconstruct the original polynomial by adding these scaled polynomials together.
This method simplifies the polynomial manipulation, leveraging orthogonal polynomials' unique properties.

Polynomial Transformation

Polynomial transformation involves modifying or expressing a given polynomial in a different basis, often to simplify calculations or solve equations. In this exercise, we transform the given polynomial \(3x^2 + x - 1\) using Legendre polynomials as the new basis.
We begin with the highest power term, which is \(3x^2\). Knowing that:
\(P_2(x) = \frac{1}{2}(3x^2 - 1)\), we isolate \(3x^2\) as \(3x^2 = 2P_2(x) + 1\).
Next, we sum up terms to match all parts of the original polynomial:
\(3x^2 + x - 1 = 2P_2(x) + P_1(x) - P_0(x)\)
This transformation simplifies solving and understanding polynomial properties by leveraging Legendre polynomials' orthogonality and simplicity.

Series Expansion

Series expansion is a powerful mathematical tool used to approximate functions by sums of simpler, known functions, often polynomials. Here, we expand the polynomial \(3x^2 + x - 1\) in terms of Legendre polynomials:
Starting from step-by-step solutions, we used:
\(3x^2 = 2P_2(x) + 1\)
The linear term \(x\) is matched with \(P_1(x)\), and the constant term is adjusted accordingly:
Ultimately, we write:
\(3x^2 + x - 1 = 2P_2(x) + P_1(x) - P_0(x)\)
In this way, each term in the series matches the form of a Legendre polynomial, allowing for easier manipulation and understanding of the original polynomial.