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. \(7 x^{4}-3 x+1\)

Short answer

\( \frac{8}{5}P_4(x) + 2P_2(x) - 3P_1(x) + \frac{2}{5}P_0(x) \)

Step-by-step solution

Write down the polynomial

The given polynomial is: \[7x^{4} - 3x + 1\]

Recall the Legendre polynomials

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

Match the highest power term

The highest power term is \(7x^4\). The coefficient of \(x^4\) in \[P_4(x)\] is \(\frac{35}{8}\). To match this, we need to solve: \[7x^4 = c \cdot \frac{35}{8}x^4 \]. This gives us our constant \(c = \frac{56}{35} = \frac{8}{5}\). Therefore, the contribution is: \[\frac{8}{5} P_4(x)\]

Subtract the highest power term contribution

Subtract \(\frac{8}{5} P_4(x)\) from the original polynomial: \[ (7x^4 - 3x + 1) - \frac{8}{5}\left(\frac{35}{8}x^4 - \,\frac{30}{8}x^2 + \frac{3}{8}\right) \]. Simplifying yields: \[- 3x + 1 + 3x^2 - \frac{3}{5}\]

Combine remaining terms with respective Legendre polynomials

Next, match the \(3x^2\) term to \(P_2(x)\) and subtract it from the remaining polynomial. Note that the coefficient of \(x^2\) in \(P_2(x)\) is \(\frac{3}{2}\). So we have: \[3x^2 = c \cdot \frac{3}{2}x^2\]. Solving for \(c\): \(c = 2\). Thus, \(3x^2 = 2P_2(x)\). Subtracting, we get: \[ -3x + 1 - \frac{3}{5} + 1 - 2 \left( \frac{3}{2} x^2 - \frac{1}{2} \right)\] and simplify.

Combine remaining linear terms

Next, we should match the linear term \(-3x\) to \(P_1(x)\). The coefficient in \(P_1(x)\) is 1. Thus, we have: \(-3x = -3P_1(x)\). Finally, subtract any remaining constant term from the remaining polynomial.

Final Result

Combining all parts: we get the linear combination: \[ \frac{8}{5}P_4(x) + 2P_2(x) - 3P_1(x) + \left( 1 - \frac{3}{5} - 1 \right)P_0(x) = \frac{8}{5} P_4(x) + 2 P_2(x) - 3 P_1(x) + \frac{2}{5} P_0(x) \].

Key concepts

Polynomial Expansion

Polynomial expansion involves writing a given polynomial as a series of terms, each multiplied by a constant and a power of the variable. In our example, we start with the polynomial: \[7x^{4} - 3x + 1\]This polynomial already has a clear expansion in terms of powers of \(x\). We will express each of these powers as a combination of Legendre polynomials which are themselves polynomials with specific properties. We do this by matching the powers of \(x\) carefully and systematically. This technique ensures that we can break down the polynomial into more manageable parts that form the basis for our solution.

Linear Combination

A linear combination means expressing a function as the sum of other functions, each multiplied by a suitable constant. For the polynomial given, \[7x^{4} - 3x + 1\],we want to express it as a linear combination of Legendre polynomials.

• We start with the highest power term and work our way down.
• In each step, we match the coefficients by solving for the constants that give the correct combination.

This involves using properties of Legendre polynomials, such as their specific coefficients at different powers of \(x\).

Orthogonal Polynomials

Orthogonal polynomials are a set of polynomials that are perpendicular to each other with respect to some inner product. Legendre polynomials are one such set. The inner product, in this case, can be described as an integral over a certain range of values.

• The orthogonality property means that the integral of the product of two different Legendre polynomials over a certain interval is zero.
• This property helps in simplifying calculations and ensures uniqueness in polynomial expansions, as each Legendre polynomial represents a different 'direction' in function space.

For our problem, using orthogonal polynomials helps in systematically decomposing the given polynomial into a sum of these orthogonal components.

Legendre Polynomial Properties

Legendre polynomials are a sequence of orthogonal polynomials with several useful properties.

• Standard form: They are usually defined with specific coefficients that depend on the polynomial's degree.
• Recursion: Each Legendre polynomial can be derived from earlier ones using a recursive relation.
• Normalization: They are normalized so that their values make them orthogonal over the interval \([-1, 1]\).

In the example, the values of \(P_0(x), P_1(x), P_2(x), P_3(x), \) and \(P_4(x)\) are crucial as they provide the types of terms that we use to reconstruct the given polynomial:

• \(P_0(x) = 1\)
• \(P_1(x) = x\)
• \(P_2(x) = \frac{1}{2}(3x^2 - 1)\)
• \(P_3(x) = \frac{1}{2}(5x^3 - 3x)\)
• \(P_4(x) = \frac{1}{8}(35x^4 - 30x^2 + 3)\)

These forms make it easier to match coefficients and solve for the constants that will express the given polynomial as a linear combination of Legendre polynomials.