Question

Show that any polynomial of degree \(n\) can be written as a linear combination of Legendre polynomials with \(l \leq n\).

Short answer

Any polynomial of degree \(n\) can be written as a sum of Legendre polynomials up to degree \(n\).

Step-by-step solution

Understanding Legendre Polynomials

Legendre polynomials, denoted as \(P_l(x)\), are a set of orthogonal polynomials. They are solutions to Legendre's differential equation and form an orthogonal basis for polynomial functions.

Representation of Polynomials

Any polynomial of degree \(n\) can be written as \(P(x) = a_0 + a_1 x + a_2 x^2 + ... + a_n x^n\). We aim to express this polynomial as a linear combination of Legendre polynomials.

Orthogonality Property

Legendre polynomials \(P_l(x)\) satisfy the orthogonality condition: \[ \int_{-1}^{1} P_m(x) P_n(x) \, dx = \frac{2}{2n + 1} \delta_{mn} \] where \(\delta_{mn}\) is the Kronecker delta.

Projection Coefficients

To express any polynomial \(P(x)\) as a linear combination of Legendre polynomials, compute the coefficients \(c_l\) using the orthogonality property: \[ c_l = \frac{2l + 1}{2} \int_{-1}^{1} P(x) P_l(x) \, dx \]

Constructing the Linear Combination

The polynomial \(P(x)\) can now be written as the sum: \[ P(x) = \sum_{l=0}^{n} c_l P_l(x) \]

Verifying the Degree

Since we only use terms up to \(l = n\), the highest degree term in the expansion will be of degree \(n\), matching the degree of the original polynomial.

Key concepts

Orthogonal Polynomials

Orthogonal polynomials are a special class of polynomials where each pair within the set is orthogonal to each other with respect to some weight function over a certain interval. For Legendre polynomials, which are a type of orthogonal polynomial, this interval is \([-1, 1]\) and the weight function is simply \(1\). The orthogonality condition for Legendre polynomials ensures that the integral of the product of any two different polynomials over this interval is zero.
Legendre polynomials are denoted by \(P_l(x)\), where \(l\) indicates the degree of the polynomial. Each \(P_l(x)\) satisfies a specific differential equation known as Legendre's differential equation.
By using orthogonal polynomials, we can simplify problems in mathematical physics and engineering, because they provide a basis that simplifies operations like integration and differentiation.

Polynomial Approximation

Polynomial approximation involves expressing complex functions as sums of simpler polynomials. When we approximate a given polynomial of degree \(n\) using Legendre polynomials, we express it as a linear combination of these orthogonal polynomials.
Any polynomial of degree \(n\) can be written in the form \(P(x) = a_0 + a_1 x + a_2 x^2 + ... + a_n x^n\). By representing this polynomial with Legendre polynomials, we use coefficients \(c_l\) for each Legendre polynomial \(P_l(x)\) from 0 to \(n\). This approach simplifies analysis by leveraging the orthogonal properties of Legendre polynomials, allowing for easier manipulation and simplification of expressions.

Orthogonality Property

The orthogonality property of Legendre polynomials is fundamental in simplifying polynomial approximation. The orthogonality condition is given by: \[ \int_{-1}^{1} P_m(x) P_n(x) \, dx = \frac{2}{2n + 1} \, \delta_{mn} \. \] Here, \delta_{mn}\ is the Kronecker delta, which is 1 when \(m = n\) and 0 otherwise.
This property allows the coefficients of the linear combination (known as projection coefficients) to be computed easily. When a function or polynomial is projected onto the basis of orthogonal polynomials, each coefficient is determined independently of the others.
To find the coefficient \(c_l\) for \(P_l(x)\), we use the formula: \ \[ c_l = \frac{2l + 1}{2} \int_{-1}^{1} P(x) P_l(x) \, dx \. \] Here, \(P(x)\) is the polynomial we want to approximate, and the integral calculates the contribution of \(P_l(x)\) to this approximation.

Legendre Differential Equation

Legendre polynomials are solutions to the Legendre differential equation, a second-order linear differential equation. This equation is: \[ \frac{d}{dx} \left( (1 - x^2) \frac{dP_l(x)}{dx} \right) + l(l + 1) P_l(x) = 0 \] Each \(P_l(x)\) for \(l = 0, 1, 2, \, ...\) is a polynomial of degree \(l\).
These polynomials arise frequently in physics, such as in solving problems involving spherical symmetry, like the Schrödinger equation for hydrogen atoms or gravitational potentials.
The existence of these polynomials and their orthogonality help address solutions in complex domains by providing a structured approach to decomposing functions into manageable components. Using the Legendre differential equation, one can generate all Legendre polynomials systematically, ensuring a comprehensive basis for polynomial approximations.