In mathematics, the Mellin transform is an integral transform that may be regarded as the multiplicative version of the two-sided Laplace transform. This integral transform is closely connected to the theory of Dirichlet series, and is often used in number theory, mathematical statistics, and the theory of asymptotic expansions; it is closely related to the Laplace transform and the Fourier transform, and the theory of the gamma function and allied special functions.
Contents
- Relationship to other transforms
- CahenMellin integral
- Number theory
- As an isometry on L2 spaces
- In probability theory
- Problems with Laplacian in cylindrical coordinate system
- Applications
- Examples
- References
The Mellin transform of a function f is
The inverse transform is
The notation implies this is a line integral taken over a vertical line in the complex plane. Conditions under which this inversion is valid are given in the Mellin inversion theorem.
The transform is named after the Finnish mathematician Hjalmar Mellin.
Relationship to other transforms
The two-sided Laplace transform may be defined in terms of the Mellin transform by
and conversely we can get the Mellin transform from the two-sided Laplace transform by
The Mellin transform may be thought of as integrating using a kernel xs with respect to the multiplicative Haar measure,
We also may define the Fourier transform in terms of the Mellin transform and vice versa; in terms of the Mellin transform and of the two-sided Laplace transform defined above
We may also reverse the process and obtain
The Mellin transform also connects the Newton series or binomial transform together with the Poisson generating function, by means of the Poisson–Mellin–Newton cycle.
The Mellin transform may also be viewed as the Gelfand transform for the convolution algebra of the locally compact abelian group of positive real numbers with multiplication.
Cahen–Mellin integral
For
where
Number theory
An important application in number theory includes the simple function
for which
assuming
As an isometry on L2 spaces
In the study of Hilbert spaces, the Mellin transform is often posed in a slightly different way. For functions in
In other words, we have set
This operator is usually denoted by just plain
Furthermore, this operator is an isometry, that is to say
In probability theory
In probability theory, the Mellin transform is an essential tool in studying the distributions of products of random variables. If X is a random variable, and X+ = max{X,0} denotes its positive part, while X − = max{−X,0} is its negative part, then the Mellin transform of X is defined as
where γ is a formal indeterminate with γ2 = 1. This transform exists for all s in some complex strip D = {s : a ≤ Re(s) ≤ b} , where a ≤ 0 ≤ b.
The Mellin transform
Problems with Laplacian in cylindrical coordinate system
In the Laplacian in cylindrical coordinates in a generic dimension (orthogonal coordinates with one angle and one radius, and the remaining lengths) there is always a term:
For example, in 2-D polar coordinates the laplacian is:
and in 3-D cylindrical coordinates the laplacian is,
This term can be easily treated with the Mellin transform, since:
For example, the 2-D Laplace equation in polar coordinates is the PDE in two variables:
and by multiplication:
with a Mellin transform on radius becomes the simple harmonic oscillator:
with general solution:
Now let's impose for example some simple wedge boundary conditions to the original Laplace equation:
these are particularly simple for Mellin transform, becoming:
These conditions imposed to the solution particularise it to:
Now by the convolution theorem for Mellin transform, the solution in the Mellin domain can be inverted:
where the following inverse transform relation was employed:
where
Applications
The Mellin Transform is widely used in computer science for the analysis of algorithms because of its scale invariance property. The magnitude of the Mellin Transform of a scaled function is identical to the magnitude of the original function for purely imaginary inputs. This scale invariance property is analogous to the Fourier Transform's shift invariance property. The magnitude of a Fourier transform of a time-shifted function is identical to the magnitude of the Fourier transform of the original function.
This property is useful in image recognition. An image of an object is easily scaled when the object is moved towards or away from the camera.
In quantum mechanics and especially quantum field theory, Fourier space is enormously useful and used extensively because momentum and position are Fourier transforms of each other (for instance, Feynman diagrams are much more easily computed in momentum space). In 2011, A. Liam Fitzpatrick, Jared Kaplan, João Penedones, Suvrat Raju, and Balt C. van Rees showed that Mellin space serves an analogous role in the context of the AdS/CFT correspondence.