Legendre 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. These numbers are, up to the factors $\frac{2n+1}{2}$, the coefficients of the Fourier–Legendre series 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.