Legendre transform (integral transform)

In mathematics, Legendre transform is an integral transform named after the mathematician Adrien-Marie Legendre, which uses Legendre polynomials $$P_n(x)$$ as kernels of the transform. Legendre transform is a special case of Jacobi transform.

The Legendre transform of a function $$f(x)$$ is


 * $$\mathcal{J}_n\{f(x)\} = \tilde f(n) = \int_{-1}^1 P_n(x)\ f(x) \ dx$$

The inverse Legendre transform is given by


 * $$\mathcal{J}_n^{-1}\{\tilde f(n)\} = f(x) = \sum_{n=0}^\infty \frac{2n+1}{2} \tilde f(n) P_n(x)$$

Associated Legendre transform
Associated Legendre transform is defined as


 * $$\mathcal{J}_{n,m}\{f(x)\} = \tilde f(n,m) = \int_{-1}^1 (1-x^2)^{-m/2}P_n^m(x) \ f(x) \ dx$$

The inverse Legendre transform is given by


 * $$\mathcal{J}_{n,m}^{-1}\{\tilde f(n,m)\} = f(x) = \sum_{n=0}^\infty \frac{2n+1}{2}\frac{(n-m)!}{(n+m)!} \tilde f(n,m)(1-x^2)^{m/2} P_n^m(x)$$