Integral transforms, operational calculus Transforms with special function kernels Theorem

Hankel inversion theorem

Builds on Hankel transform Bessel function of the first kind

Statement

Let $\nu \ge -\frac12$ , and let $J_\nu$ be the Bessel function of the first kind of order $\nu$ . Let $f \colon (0, \infty) \to \C$ satisfy $\int_0^\infty \abs{f(r)}\, r^{1/2}\, dr < \infty$ , and let $F_\nu(k) = \int_0^\infty f(r) J_\nu(kr)\, r\, dr$ be its Hankel transform of order $\nu$ . Let $r > 0$ , and assume that $f$ is of bounded variation on a neighbourhood of $r$ . Then $\lim_{K \to \infty} \int_0^K F_\nu(k)\, J_\nu(kr)\, k\, dk = \frac{f(r^+) + f(r^-)}{2}.$ In particular, the limit is $f(r)$ at every point $r$ where $f$ is continuous.

Remarks

The proof follows the pattern of the Fourier inversion theorem for Fourier integrals. After exchanging the integrals, the inner integral $\int_0^K J_\nu(kr) J_\nu(kt)\, k\, dk$ is computed in closed form from Bessel's equation, and it acts as an approximate identity concentrated at $t = r$ as $K \to \infty$ . A Riemann–Lebesgue argument disposes of the parts away from $r$ , and the bounded variation near $r$ controls the part near $r$ , exactly as in Dirichlet's theorem for Fourier series. There is also an $L^2$ version: the Hankel transform of order $\nu$ extends to a unitary operator of $L^2((0, \infty), r\, dr)$ that is its own inverse.

Example: for $\nu = 0$ and $f(r) = e^{-r^2/2}$ one has $F_0(k) = e^{-k^2/2}$ , and transforming again gives back $e^{-r^2/2}$ . For the indicator function of $[0, 1]$ and $\nu = 0$ , $F_0(k) = J_1(k)/k$ , and at $r = 1$ the inversion integral converges to the mean value $\frac12$ of the jump.

Going deeper

For a student

The theorem is the radial form of the Fourier inversion theorem, and for $\nu = \frac{n}{2} - 1$ it can be derived from it for radial functions on $\R^n$ . It justifies the solution of boundary value problems in cylindrical coordinates: transform in $r$ , solve an ordinary differential equation in the remaining variable, and transform back. The hypotheses parallel those of Dirichlet's pointwise convergence theorem for Fourier series.

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Bessel function of the first kind concept
Hankel transform concept
Hankel inversion theorem statement

Tag	`BMXZSXETDW`
URI	`m2://BMXZSXETDW`
Kind	statement, theorem
Writing	W0 stub
MSC	44A20, 33C10
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About	Hankel transform
Uses	Hankel transform, Bessel function of the first kind