Hankel Transform - The Plancherel Theorem and Parseval's Theorem

The Plancherel Theorem and Parseval's Theorem

If f(r) and g(r) are such that their Hankel transforms Fν(k) and Gν(k) are well defined, then the Plancherel theorem states

\int_0^\infty f(r)g(r)r~\operatorname{d}r = \int_0^\infty F_\nu(k)G_\nu(k) k~\operatorname{d}k.

Parseval's theorem, which states:

\int_0^\infty |f(r)|^2r~\operatorname{d}r = \int_0^\infty |F_\nu(k)|^2 k~\operatorname{d}k.

is a special case of the Plancherel theorem. These theorems can be proven using the orthogonality property.

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