---
tag: 51PH7Z03TJ
kind: concept
name: Legendre integral transform
msc: [44A15, 42C10, 33C45]
specializes: [YCK709QB06]
sources:
  - url: https://en.wikipedia.org/wiki/Legendre_transform_(integral_transform)
same_as:
  wikidata: Q30587593
  wikipedia: Legendre transform (integral transform)
---

## Summary

The Legendre integral transform sends a function $f$ on $[-1, 1]$ to the sequence of numbers $\int_{-1}^{1} P_n(x) f(x)\, dx$, where $P_n$ is the $n$-th [[Legendre polynomial|QFYGJVA996]].
These numbers are, up to the factors $\frac{2n+1}{2}$, the coefficients of the [[Fourier–Legendre series|RK2G9HN63E]] of $f$, so the series is the inverse transform.
It plays for the Legendre differential operator the role that Fourier coefficients play for $d^2/dx^2$.
It turns Legendre's differential equation into an algebraic equation.
It should not be confused with the Legendre transformation of convex analysis.

## Explanations

### L1 human

The Legendre polynomials $1, x, \frac{3x^2 - 1}{2}, \dots$ are like a set of perpendicular directions for functions on $[-1, 1]$.
The transform measures how much of each direction a function contains.
For example, the function $f(x) = x$ is exactly $P_1$, so its transform is $\frac{2}{3}$ at $n = 1$ and $0$ at every other $n$.