Question

Show that any polynomial of degree $n$ can be written as a linear combination of Legendre polynomials with $l\le n$

Short answer

A polynomial of degree $n$ can be expressed as a linear combination of Legendre polynomials of degrees $l$ with $l\le n$ using orthogonal properties and integration.

Step-by-step solution

Understand Legendre Polynomials

Legendre polynomials are a set of orthogonal polynomials which arise in solving certain types of differential equations. They are denoted as $P_l(x)$ where $l$ is a non-negative integer.

Define the Polynomial

Consider a polynomial $P(x)$ of degree $n$, which can be written as $$P(x)=a_0+a_{1}x+a_2{x}^2+...+a_n{x}^n$$

Express Polynomial as a Linear Combination

To show that $P(x)$ can be expressed as a linear combination of Legendre polynomials, assume $$P(x)=c_0{P}_0(x)+c_1{P}_1(x)+c_2{P}_2(x)+...+c_n{P}_n(x)$$ where $c_i$ are constants and $P_i(x)$ are Legendre polynomials of degree $i$.

Use Orthogonality

Legendre polynomials are orthogonal, meaning $${\int }_{-1}^1{P}_i(x)P_j(x)dx=0\text{for}ieqj$$. This orthogonality helps in finding the coefficients $c_i$.

Compute the Coefficients

Multiply both sides of the polynomial equation by $P_k(x)$ for a fixed $k$ and integrate over the interval $[-1,1]$. $${\int }_{-1}^{1}P(x)P_k(x)dx={\int }_{-1}^1(\sum _{i=0}^n{c}_i{P}_i(x))P_k(x)dx$$ Due to orthogonality, all terms where $ieqk$ vanish, leaving $${\int }_{-1}^{1}P(x)P_k(x)dx=c_k{\int }_{-1}^1(P_k(x){)}^{2}dx$$

Solve for the Constants

Solve for $c_k$ by isolating it: $$c_k=\frac{{\int }_{-1}^{1}P(x)P_k(x)dx}{{\int }_{-1}^1(P_k(x){)}^{2}dx}$$

Conclusion

Using the calculated coefficients $c_k$, substitute them back into the linear combination, showing that the polynomial of degree $n$ can be written as a sum of Legendre polynomials with degrees up to $n$.

Key concepts

Orthogonal Polynomials

Orthogonal polynomials, like Legendre polynomials, have unique properties that make them useful in many fields of study such as physics and engineering.
These polynomials are 'orthogonal' over a specified interval with respect to a weight function. This means that their inner product is zero when integrating the product of two different polynomials over that interval.
For Legendre polynomials, this interval is $[-1,1]$ and the weight function is 1.
Orthogonality simplifies many calculations, especially when dealing with polynomial approximations.
Because of orthogonality, it becomes easier to decompose complex functions into simpler, orthogonal components.

Polynomial Decomposition

Polynomial decomposition involves breaking down a complex polynomial into simpler components, often a linear combination of simpler polynomials.
This is particularly useful when analyzing the behavior of the polynomial or solving equations.
In the context of Legendre polynomials, any polynomial of degree $n$ can be expressed as $$P(x)=c_0{P}_0(x)+c_1{P}_1(x)+c_2{P}_2(x)+...+c_n{P}_n(x)$$
Here, $c_i$ are coefficients, and $P_i(x)$ are Legendre polynomials of degree $i$.
This decomposition makes it easier to handle complex polynomials by leveraging the orthogonal characteristics of Legendre polynomials.

Coefficients Calculation

Calculating the coefficients in the polynomial decomposition is an essential step.
For this, we use the orthogonality property of Legendre polynomials.
To find a specific coefficient, say $c_k$, we multiply the entire polynomial by $P_k(x)$ and integrate over the interval $[-1,1]$.
Due to orthogonality, the integral simplifies significantly, resulting in the formula: $$c_k=\frac{{\int }_{-1}^{1}P(x)P_k(x)dx}{{\int }_{-1}^1(P_k(x){)}^{2}dx}$$
This formula isolates $c_k$, allowing us to determine the coefficients needed for the polynomial decomposition.

Orthogonality in Polynomials

Orthogonality in polynomials provides a powerful tool for polynomial analysis and decomposition.
When polynomials are orthogonal, as with Legendre polynomials, it simplifies many integral calculations.
The orthogonality condition, $${\int }_{-1}^1{P}_i(x)P_j(x)dx=0\text{for}iej,$$
means that the integral of the product of two different orthogonal polynomials over the interval $[-1,1]$ is zero.
This property is extremely useful in calculating coefficients and simplifies the process of expressing complex polynomials as linear combinations of orthogonal polynomials.