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 $f(x)$ isChurchill, R. V., and C. L. Dolph. "Inverse transforms of products of Legendre transforms." Proceedings of the American Mathematical Society 5.1 (1954): 93–100.

:$\mathcal_n\ = \tilde f(n) = \int_^1 P_n(x)\ f(x) \ dx$

The inverse Legendre transform is given by

:$\mathcal_n^\ = f(x) = \sum_^\infty \frac \tilde f(n) P_n(x)$

Associated Legendre transform

Associated Legendre transform is defined as

:$\mathcal_\ = \tilde f(n,m) = \int_^1 (1-x^2)^P_n^m(x) \ f(x) \ dx$

The inverse Legendre transform is given by

:$\mathcal_^\ = f(x) = \sum_^\infty \frac\frac \tilde f(n,m)(1-x^2)^ P_n^m(x)$

Some Legendre transform pairs

References

Integral transforms
Mathematical physics

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