Hankel transform

The Hankel transform is in the functional analysis, a branch of mathematics, a linear integral transformation, which is essentially based on the Bessel functions of the first kind. It is named after the mathematician Hermann Hankel. Applications are, among others, in the field of image processing for correction of aberrations.

5.1 Distribution Space
5.2 Hankel transform

6.1 The hyperbola 1 / t
6.2 The bell curve
6.3 The delta function

Definition

The Hankel transform, there are different conventions to define it. Let be a complex-valued function and. Then you can see the Hankel transform of order of by

Define, here are the

Bessel functions of the first kind and is the gamma function. In this respect, the integral exists, is called the Hankel transform of. This convention is the Hankel transform is mainly used in this article. In some sections, however, the variant shown below is used, as pointed out in the relevant sections.

Another way to define the Hankel transform of order of being,

Here, the Bessel functions of the first kind with also called and is called here Hankel transform, insofar exists the integral.

Inverse Hankel transform

Similar to the Fourier transform, it is also in the Hankel transform in certain circumstances possible to recover from the Hankel transform their original function. An important result from the theory of Hankel transform states that if a Lebesgue integrable function with bounded variation, the output function from the Hankel transform with the inverse integral transform

Can be recovered. The Hankel transform and its inverse transform are therefore equal. It can thus be understood as involutive figure. For the alternative definition, this statement is analogous.

Properties

Orthogonality

The Bessel functions form an orthogonal basis: it is

For and greater than 0 and as the delta distribution.

Algebraization of the Bessel differential operator

The Bessel differential operator. So For the Bessel functions. Using the Hankel transform, it is possible to convert these differential operator in an expression without derivatives. Precise applies

This is a central feature of the Hankel transform for solving differential equations.

Relation to the Fourier transform

The Hankel transform has several analogies to the Fourier transform. In particular, can be calculated by the Hankel transform of a two-dimensional Fourier transform. For this purpose let a radially symmetric function. This means that the function is independent of, which is why it is listed in the following only with the parameter. This function will now be described with reference to the function and the Fourier transform of the Hankel transform.

To see this, the Fourier integral is

Transformed from polar coordinates, leading to

Leads. This indicates that a Fourier transform of a radially symmetrical function always corresponds to the Hankel transform of a corresponding function. In particular, it is possible to construct a given function a corresponding radially symmetric function, with which one can calculate the Hankel transform of Fourier transform.

Hankel transform of distributions

Also as in the Fourier transform, it is possible in an analogous manner in the Hankel transform, to generalize to distributions. In contrast to the Fourier transform of the Hankel transforms can not be defined on the space of tempered distributions. Therefore, we define a new space and explains the Hankel transform for distributions on its dual space.

Distribution space

Be then is defined by

On this vector space topology is also defined in terms of a convergence concept. A sequence converges to zero when

Applies to all. By forming the topological dual space we obtain the distribution space, on which one can define the Hankel transform. For example, all distributions with compact support in how the delta function is included in the room.

Hankel transform

For the Hankel transform for all defined by

The expression is again a Hankel transform of a function, and therefore defined. Due to the design of the space, the Convention for the transformation is used here, however.

As for the Fourier transformation is carried distributions for the Hankel transform of the distribution is not even, but rather it is calculated on the test function.

Examples

This section of the Bessel functions of the second kind of order n, with the gamma function, with the imaginary unit and again denotes the delta distribution. In the table on the right side of some pairs of Hankel transforms are additionally listed.

The hyperbola 1 / t

For the Hankel transform of zero order of

So the function is a fixed point of the Hankel transform.

The bell curve

In this section the calculation of the Hankel transform of the Gaussian bell curve is outlined with the help of the Fourier transform. Since the function is analytically, it may be continued on and there is even radially symmetrical. Therefore, the Hankel transform may be calculated using the Fourier transform on. The Fourier transform is a fixed point, from which it follows that the Hankel transform of also again. So the Gaussian bell curve is also a fixed point of the Hankel transform.

The delta function

In this example, the Hankel transform of zero order of the delta distribution is calculated. It is

The term is to be understood as a distribution that is generated by the constant one function. In the field of physics, we listed the delta distribution often imprecise as a real-valued function and not as functional. In this case, the calculation of the Hankel transform shortened to

If you want to inversely calculate the Hankel transform of the constant one function, you come upon insertion into the integral representation of a divergent integral. Because of tightness arguments, it is still possible to understand the delta function as a Hankel transform of the constant function one.

Swell

Larry C. Andrews, Bhimsen K. Shivamoggi: Integral Transforms for Engineers. SPIE Press, University of Central Florida, 1999, ISBN 978-0-81943232-2, Chapter 7

Alexander D. Poularikas: The Transforms and Applications Handbook. 2nd edition. CRC Press, 2000, ISBN 978-0849385957, Chapter 9

LS Dube, JN Pandey: On the Hankel transform of distributions Tohoku Math J. ( 2) Volume 27, Number 3 ( 1975), 337-354.

Involution (mathematics) Differentialoperator Topological space Analytic function Functional (mathematics) Wolfram Research

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