---
tag: T046A1477B
kind: definition
name: Fourier–Legendre series
msc: [42C10, 33C45]
defines: RK2G9HN63E
uses: [QFYGJVA996, SASP14MA1Z, DWJF7ABPR1]
sources:
  - url: https://en.wikipedia.org/wiki/Legendre_polynomials
---

## Definition

Let $P_n$ be the [[Legendre polynomial|QFYGJVA996]] of degree $n$, for $n \ge 0$, normalized by $P_n(1) = 1$.
Let $f \colon [-1, 1] \to \C$ be a [[Lebesgue integrable|DWJF7ABPR1]] function.
The *Fourier–Legendre coefficients* of $f$ are
$$a_n = \frac{2n + 1}{2} \int_{-1}^{1} f(x) P_n(x) \, dx, \qquad n \ge 0.$$
The *Fourier–Legendre series* of $f$ is the formal series
$$\sum_{n=0}^{\infty} a_n P_n(x).$$

## Remarks

The factor $(2n+1)/2$ comes from the orthogonality relations $\int_{-1}^{1} P_m P_n \, dx = \frac{2}{2n+1} \delta_{mn}$.
For $f$ in the [[space|SASP14MA1Z]] $L^2([-1, 1])$, the series converges to $f$ in $L^2$, because the Legendre polynomials form a complete orthogonal system there.

Example: let $f(x) = x^2$.
Then $a_0 = \frac12 \int_{-1}^1 x^2 \, dx = \frac13$, $a_1 = \frac32 \int_{-1}^1 x^3 \, dx = 0$, and $a_2 = \frac52 \int_{-1}^1 x^2 \cdot \frac{3x^2 - 1}{2} \, dx = \frac52 \cdot \frac{4}{15} = \frac23$.
All $a_n$ with $n \ge 3$ vanish, since $x^2$ is a combination of $P_0$ and $P_2$ and is orthogonal to $P_n$ for $n \ge 3$.
So the series is $\frac13 P_0 + \frac23 P_2$, which equals $x^2$.

Non-example: $f(x) = 1/(1 - x)$ on $[-1, 1)$ is not integrable near $x = 1$.
Already $a_0 = \frac12 \int_{-1}^{1} \frac{dx}{1 - x}$ diverges, so $f$ has no Fourier–Legendre series.