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. $$5-2x$$

Short answer

5 - 2x = 5P_0(x) - 2P_1(x)

Step-by-step solution

Write Down Legendre Polynomials

The first few Legendre polynomials are: $$P_0(x)=1$$ $$P_1(x)=x$$ Since we only need up to the first-order polynomials for this problem, these are sufficient.

Express Given Polynomial as Linear Combination

We need to express the given polynomial, $5-2x$, as a linear combination of Legendre polynomials. Let’s assume: $$5-2x=a_0{P}_0(x)+a_1{P}_1(x)$$ where $a_0$ and $a_1$ are coefficients to be determined.

Match Coefficients

To find the coefficients $a_0$ and $a_1$, we compare the terms on both sides. Since $$5-2x=a_0\times 1+a_1\times x$$ It is clear from this comparison that: $$a_0=5$$ and $$a_1=-2$$

Write the Final Linear Combination

Substituting the values of $a_0$ and $a_1$ back into the equation, we get: $$5-2x=5P_0(x)-2P_1(x)$$

Key concepts

linear combination

In mathematics, a linear combination involves adding together different elements, each multiplied by a coefficient or constant. When dealing with polynomials, a linear combination means combining multiple polynomials in such a way that their sum gives another polynomial. This is fundamental in expressing functions in different bases.

In the given problem, we have the polynomial $5-2x$. This can be expressed as a linear combination of the Legendre polynomials $P_0(x)=1$ and $P_1(x)=x$. We write the polynomial as:
$5-2x=a_0{P}_0(x)+a_1{P}_1(x)$. Here, $a_0$ and $a_1$ are the coefficients that we need to find.

By matching coefficients, we can determine the values of $a_0$ and $a_1$. This gives us
$a_0=5$ and $a_1=-2$. Hence, the polynomial $5-2x$ is written as a linear combination of Legendre polynomials:$5-2x=5P_0(x)-2P_1(x)$.

orthogonal polynomials

Orthogonal polynomials are a sequence of polynomials where each polynomial is orthogonal to every other polynomial in the sequence with respect to a given inner product. This orthogonality property means that the integral of the product of any two different polynomials over a specified range is zero.

The Legendre polynomials $P_n(x)$ form an example of such an orthogonal polynomial sequence. For Legendre polynomials, the orthogonality condition is:

$$otag{\int }_{-1}^1{P}_m(x)P_n(x)dx=0\text{if}meqn$$

This property is very useful for simplifying problems in mathematical physics and engineering, especially in solving differential equations. The orthogonality helps in decomposing a given function into a series of polynomials that can be individually analyzed.

polynomial decomposition

Polynomial decomposition refers to the process of breaking a polynomial into a sum of simpler polynomials. This can often make problems easier to solve or analyze.

For example, in the problem, the polynomial $5-2x$ was decomposed into the sum of the simpler Legendre polynomials: $5P_0(x)-2P_1(x)$.

This technique leverages the properties of the basis polynomials (like orthogonality in the case of Legendre polynomials) to simplify calculations. Once decomposed, each term in the sum can be treated individually, which helps in solving equations, integrating, or differentiating the polynomial.