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Wikipedia / Mellin transform

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 essentially the same as the Laplace transform and closely related to the Fourier transform, and the theory of the gamma function and allied special functions.

The Mellin transform of a complex-valued function given (where it exists, see is a Haar measure is a (in general non-unitary) multiplicative character The notation implies this is a line integral taken over a vertical line in the complex plane, whose real part c need only satisfy a mild lower bound. Conditions under which this inversion is valid are given in the Mellin inversion theorem.

The transform was introduced in 1859 by Bernhard Riemann. The transform is named after the Finnish mathematician Hjalmar Mellin, who developed the first rigorous account in paper published 1897 in Acta Societatis Scientiarum Fennicae.

Relationship to other transforms

The two-sided Laplace transform

The Mellin transform may be thought of as integrating using a kernel with respect to the multiplicative Haar measure the two-sided Laplace transform integrates with respect to the additive Haar measure , which is translation invariant, so that .

We also may define the Fourier transform

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.

Examples

Cahen–Mellin integral

is the gamma function is a meromorphic function with simple poles at . is analytic for on the principal branch

This integral is known as the Cahen–Mellin integral.

Polynomial functions

is not convergent for any value of

and is thus defined for and is thus defined for .

Exponential functions

For , let

Zeta function

It is possible to use the Mellin transform to produce one of the fundamental formulas for the Riemann zeta function, . Let

Generalized Gaussian

For is a generalized Gaussian distribution

Power series and Dirichlet series

Generally, assuming the necessary convergence, we can connect Dirichlet series and power series by this formal identity involving the Mellin transform:

Fundamental strip

For with . The fundamental strip is define its right endpoint. To summarize using Big O notation as is defined in the strip .

An application of this can be seen in the gamma function, for all should be defined in the strip is analytic for .

Properties

The properties in this table may be found in and .

Parseval's theorem and Plancherel's theorem

in the fundamental strips with are also square-integrable over the interval , then Parseval's formula holds: that lies entirely within the overlap of the (suitable transformed) fundamental strips.

by in the fundamental strips with , then we have by in the fundamental strip with is also square-integrable over the interval , then Plancherel's theorem holds:

As an isometry on L2 spaces

In the study of Hilbert spaces (see Lp space) the fundamental strip always includes , so we may define a linear operator is used here to distinguish from the definition used elsewhere in this article. The Mellin inversion theorem Furthermore, this operator is an isometry was used).

In probability theory

In probability theory, the Mellin transform is an essential tool in studying the distributions of products of random variables. If is a random variable, and denotes its positive part, while is its negative part, then the Mellin transform of is defined as where is a formal indeterminate with . This transform exists for all in some complex strip , where .

of a random variable uniquely determines its distribution function . The importance of the Mellin transform in probability theory lies in the fact that if and are two independent random variables, then the Mellin transform of their product is equal to the product of the Mellin transforms of and :

Problems with Laplacian in cylindrical coordinate system

In the Laplacian in cylindrical coordinates in a generic dimension (orthogonal coordinates

This term can be treated with the Mellin transform,

For example, the 2-D Laplace equation with a Mellin transform on radius becomes the simple harmonic oscillator

Now let's impose for example some simple wedge boundary conditions

.

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.

Examples

Table of selected Mellin transforms

Below is a list of interesting examples for the Mellin transform:

See also

Notes

References