Mellin transform
Encyclopedia
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
and the theory of asymptotic expansion
s; it is closely related to the Laplace transform and the Fourier transform
, and the theory of the gamma function
and allied special functions.
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
.
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
,
, which is invariant
under dilation
, so that
; 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
in terms of the Mellin transform and vice-versa; if we define the two-sided Laplace transform as above, then
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.
, 
and 
on the principal branch
, one has
where
is the gamma function
. This integral is known as the Cahen-Mellin integral.
for which
s, the Mellin transform is often posed in a slightly different way. For functions in
(see Lp space
) the fundamental strip always includes
, so we may define a linear operator 
as
In other words we have set
This operator is usually denoted by just plain
and called the "Mellin transform", but 
is used here to distinguish from the definition used elsewhere in this article. The Mellin inversion theorem
then shows that
is invertible with inverse
Furthermore this operator is an isometry
, that is to say
for all 
(this explains why the factor of 
was used). Thus 
is a unitary operator
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Mellin transform is an integral transform that may be regarded as the multiplicative
Multiplicative group
In mathematics and group theory the term multiplicative group refers to one of the following concepts, depending on the context*any group \scriptstyle\mathfrak \,\! whose binary operation is written in multiplicative notation ,*the underlying group under multiplication of the invertible elements of...
version of the two-sided Laplace transform
Two-sided Laplace transform
In mathematics, the two-sided Laplace transform or bilateral Laplace transform is an integral transform closely related to the Fourier transform, the Mellin transform, and the ordinary or one-sided Laplace transform...
. This integral transform is closely connected to the theory of Dirichlet series, and is
often used in number theory
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
and the theory of asymptotic expansion
Asymptotic expansion
In mathematics an asymptotic expansion, asymptotic series or Poincaré expansion is a formal series of functions which has the property that truncating the series after a finite number of terms provides an approximation to a given function as the argument of the function tends towards a particular,...
s; it is closely related to the Laplace transform and the Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
, and the theory of the gamma function
Gamma function
In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
and allied special functions.
The Mellin transform of a function f is
The inverse transform is
The notation implies this is a line integral
Line integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve.The function to be integrated may be a scalar field or a vector field...
taken over a vertical line in the complex plane. Conditions under which this inversion is valid are given in the Mellin inversion theorem
Mellin inversion theorem
In mathematics, the Mellin inversion formula tells us conditions underwhich the inverse Mellin transform, or equivalently the inverse two-sided Laplace transform, are defined and recover the transformed function....
.
The transform is named after the Finnish
Finland
Finland , officially the Republic of Finland, is a Nordic country situated in the Fennoscandian region of Northern Europe. It is bordered by Sweden in the west, Norway in the north and Russia in the east, while Estonia lies to its south across the Gulf of Finland.Around 5.4 million people reside...
mathematician Hjalmar Mellin
Hjalmar Mellin
Robert Hjalmar Mellin was a Finnish mathematician and functional theorist.He studied at the University of Helsinki and later in Berlin under Karl Weierstrass. He is most noted as the developer of the integral transform known as the Mellin transform...
.
Relationship to other transforms
The two-sided Laplace transformTwo-sided Laplace transform
In mathematics, the two-sided Laplace transform or bilateral Laplace transform is an integral transform closely related to the Fourier transform, the Mellin transform, and the ordinary or one-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
Haar measure
In mathematical analysis, the Haar measure is a way to assign an "invariant volume" to subsets of locally compact topological groups and subsequently define an integral for functions on those groups....
,
, which is invariantunder dilation
, so that; 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
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
in terms of the Mellin transform and vice-versa; if we define the two-sided Laplace transform as above, then
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.
Cahen-Mellin integral
For, and on the principal branchPrincipal branch
In mathematics, a principal branch is a function which selects one branch, or "slice", of a multi-valued function. Most often, this applies to functions defined on the complex plane: see branch cut....
, one has
where
is the gamma functionGamma function
In mathematics, the gamma function is an extension of the factorial function, with its argument shifted down by 1, to real and complex numbers...
. This integral is known as the Cahen-Mellin integral.
Number Theory
An important application in number theory includes the simple functionfor which
As a unitary operator on L2
In the study of Hilbert spaceHilbert space
The mathematical concept of a Hilbert space, named after David Hilbert, generalizes the notion of Euclidean space. It extends the methods of vector algebra and calculus from the two-dimensional Euclidean plane and three-dimensional space to spaces with any finite or infinite number of dimensions...
s, the Mellin transform is often posed in a slightly different way. For functions in
(see Lp spaceLp space
In mathematics, the Lp spaces are function spaces defined using a natural generalization of the p-norm for finite-dimensional vector spaces...
) the fundamental strip always includes
, so we may define a linear operator asIn other words we have set
This operator is usually denoted by just plain
and called the "Mellin transform", but is used here to distinguish from the definition used elsewhere in this article. The Mellin inversion theoremMellin inversion theorem
In mathematics, the Mellin inversion formula tells us conditions underwhich the inverse Mellin transform, or equivalently the inverse two-sided Laplace transform, are defined and recover the transformed function....
then shows that
is invertible with inverseFurthermore this operator is an isometry
Isometry
In mathematics, an isometry is a distance-preserving map between metric spaces. Geometric figures which can be related by an isometry are called congruent.Isometries are often used in constructions where one space is embedded in another space...
, that is to say
for all (this explains why the factor of was used). Thus is a unitary operatorUnitary operator
In functional analysis, a branch of mathematics, a unitary operator is a bounded linear operator U : H → H on a Hilbert space H satisfyingU^*U=UU^*=I...
.
In probability theory
In probability theory Mellin transform is an essential tool in studying the distributions of products of random variables. If X is a random variable, and } denotes its positive part, while } is its negative part, then the Mellin transform of X is defined as-
where γ is a formal indeterminate with . This transform exists for all s in some complex strip }, where .
The Mellin transform
of a random variable X uniquely determines its distribution function FX. The importance of the Mellin transform in probability theory lies in the fact that if X and Y are two independent random variables, then the Mellin transform of their products is equal to the product of the Mellin transforms of X and Y:
-
Applications
The Mellin Transform is widely used in computer science 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. 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 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.
Examples
- Perron's formulaPerron's formulaIn mathematics, and more particularly in analytic number theory, Perron's formula is a formula due to Oskar Perron to calculate the sum of an arithmetical function, by means of an inverse Mellin transform.-Statement:...
describes the inverse Mellin transform applied to a Dirichlet series. - The Mellin transform is used in analysis of the prime-counting function and occurs in discussions of the Riemann zeta function.
- Inverse Mellin transforms commonly occur in Riesz meanRiesz meanIn mathematics, the Riesz mean is a certain mean of the terms in a series. They were introduced by Marcel Riesz in 1911 as an improvement over the Cesàro mean...
s.
External links
- Philippe Flajolet, Xavier Gourdon, Philippe Dumas, Mellin Transforms and Asymptotics: Harmonic sums.
- Antonio Gonzáles, Marko Riedel Celebrando un clásico, newsgroup es.ciencia.matematicas
- Juan Sacerdoti, Funciones Eulerianas (in Spanish).
- Perron's formula
-