. Orthogonality The Bessel functions form an orthogonal basis with respect to the weighting factor r: The Plancherel theorem and Parseval's theorem If f(r) and g(r) are such that their Hankel transforms and are well defined, then the Plancherel theorem states Parseval's theorem, which states is a special case of the Plancherel theorem. These theorems can be proven using the orthogonality property. Relation to the multidimensional Fourier transform The Hankel transform appears when one writes the multidimensional Fourier transform in hyperspherical coordinates, which is the reason why the Hankel transform often appears in physical problems with cylindrical or spherical symmetry. Consider a function of a -dimensional vector . Its -dimensional Fourier transform is defined asTo rewrite it in hyperspherical coordinates, we can use the decomposition of a plane wave into -dimensional hyperspherical harmonics :where and are the sets of all hyperspherical angles in the -space and -space. This gives the following expression for the -dimensional Fourier transform in hyperspherical coordinates:If we expand and in hyperspherical harmonics:the Fourier transform in hyperspherical coordinates simplifies toThis means that functions with angular dependence in form of a hyperspherical harmonic retain it upon the multidimensional Fourier transform, while the radial part undergoes the Hankel transform (up to some extra factors like ). Special cases Fourier transform in two dimensions If a two-dimensional function is expanded in a multipole series, then its two-dimensional Fourier transform is given bywhereis the -th order Hankel transform of (in this case plays the role of the angular momentum, which was denoted by in the previous section). Fourier transform in three dimensions If a three-dimensional function is expanded in a multipole series over spherical harmonics, then its three-dimensional Fourier transform is given bywhereis the Hankel transform of of order . This kind of Hankel transform of half-integer order is also known as the spherical Bessel transform. Fourier transform in dimensions (radially symmetric case) If a -dimensional function does not depend on angular coordinates, then its -dimensional Fourier transform also does not depend on angular coordinates and is given bywhich is the Hankel transform of of order up to a factor of . 2D functions inside a limited radius If a two-dimensional function is expanded in a multipole series and the expansion coefficients are sufficiently smooth near the origin and zero outside a radius , the radial part may be expanded into a power series of : such that the two-dimensional Fourier