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. \(x^{4}\)

Short answer

The linear combination is \( \frac{8}{35} P_4(x) - \frac{16}{35} P_2(x) + \frac{8}{35} \).

Step-by-step solution

Identify the Legendre Polynomials

The first few Legendre polynomials are: o \(P_0(x) = 1\) o \(P_1(x) = x\) o \(P_2(x) = \frac{1}{2}(3x^2 - 1)\) o \(P_3(x) = \frac{1}{2}(5x^3 - 3x)\) o \(P_4(x) = \frac{1}{8}(35x^4 - 30x^2 + 3)\)

Match the Highest Degree Term

Start with the highest power of \(x\) in \(P_4(x)\). Given that \(P_4(x) = \frac{1}{8}(35x^4 - 30x^2 + 3)\), we see that \(35x^4\) is the term involving \(x^4\).Thus, \(x^4\) can be written as a combination involving \(P_4(x)\). Rewrite \(x^4\) in terms of \(P_4(x)\): $$x^4 = \frac{8}{35}P_4(x) + \text{(terms without } x^4)$$

Remove Lower Degree Terms

Next, factor out the terms involving \(x^4\): $$x^4 = \frac{8}{35}(35x^4 - 30x^2 + 3) - \frac{8}{35}(-30x^2 + 3)$$That simplifies to: $$x^4 = P_4(x) - \frac{24}{35}x^2 + \frac{24}{35}$$

Express Remaining Terms Using Legendre Polynomials

The remaining terms \(- \frac{24}{35}x^2\) and \(\frac{24}{35}\) can be expressed using lower degree Legendre polynomials:- \(x^2\) can be expressed as \(P_2(x)\): $$x^2 = \frac{2}{3} P_2(x) + \frac{1}{3} $$ which rearranges to $$P_2(x) = \frac{3}{2}(x^2 - \frac{1}{3})$$Using this identity, $$- \frac{24}{35}x^2 = - \frac{24}{35} \left( \frac{2}{3} P_2(x) + \frac{1}{3} \right)$$- The constant term \(\frac{24}{35}\) can be expressed as \(P_0(x)\).

Combine All Terms

By combining all the above steps, we get:$$x^4 = \frac{8}{35}P_4(x) + \frac{24}{35}\left(- \frac{2}{3}P_2(x) - \frac{1}{3}\right) + \frac{24}{35}P_0(x)$$Simplify the coefficients and combine the terms:

Simplified Linear Combination

By simplifying, we arrive at:$$x^4 = \frac{8}{35} P_4(x) - \frac{16}{35} P_2(x) + \frac{8}{35}$$

Key concepts

Polynomial Expansion

Polynomial expansion is the process of expressing a polynomial as a sum of terms, typically using powers of a variable like x. In the case of the exercise provided, we specifically use Legendre polynomials for this expansion.
This means we are transforming a polynomial such as \(x^4\) into a linear combination of specified orthogonal polynomials.

Legendre polynomials are particularly useful because they form an orthogonal basis set, meaning any polynomial can be projected onto them without overlap. The process involves identifying the relevant Legendre polynomials that match the powers of x in our original polynomial and then combining them with appropriate coefficients.

To start, we identify specific Legendre polynomials, such as \(P_0(x) = 1\), \(P_1(x) = x\), and so on. The next step is to match these polynomials to the highest power term in our polynomial and work downwards, ensuring all terms are accounted for in the expansion.
This method ensures every term is expressed in terms of the orthogonal Legendre polynomials, allowing a thorough polynomial decomposition.

Orthogonal Polynomials

Orthogonal polynomials are a set of polynomials where each pair is orthogonal under some inner product, meaning their inner product (or integral) over a specified range equals zero when they are different. This attribute makes them powerful tools in mathematics.

Legendre polynomials are a prime example of orthogonal polynomials. They are widely used because of their properties and the weight function is constant over the interval \([-1, 1]\). Orthogonality provides the convenience that each polynomial in the set effectively projects only onto itself, minimizing cross-interference.

When we say a polynomial can be written as a linear combination of orthogonal polynomials, we mean we can represent it uniquely in terms of these basis polynomials. This transformation simplifies many problems, especially in areas like approximation theory, numerical integration, and solving differential equations.
The orthogonality of Legendre polynomials ensures that the coefficients used in the combination are optimal and unique. This is evident in the exercise, where expansion coefficients are systematically derived to match the original polynomial accurately.

Mathematical Methods

Mathematical methods refer to the standardized techniques used to solve problems and manipulate mathematical expressions. In the context of polynomial expansions, these methods involve a procedural approach to decompose and recompose polynomials.

One major method used in the exercise is matching and projecting polynomials onto a set of basis functions - here, the Legendre polynomials. This method is essential in linear algebra, calculus, and differential equations.

For instance, starting with the highest power term ensures we address the most significant component of the polynomial first. This systematic approach helps simplify the problem progressively. Breaking down terms step-by-step ensures that each component adheres to the properties of orthogonal polynomials.
Further, using identities like transforming \(x^2\) into terms of Legendre polynomials simplifies complicated expressions and helps in the final linear combination.

This structured approach means we can use familiar polynomial identities and algebraic manipulation to achieve a rigorous and clear solution. The mathematical methods showcased in the exercise highlight the importance of systematic problem-solving and decomposition in mastering complex polynomial expansions.