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 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 is
The inverse transform is
The notation implies this is a line integral taken over a vertical line in the complex plane, whose real part "c" is arbitrary provided that it meets certain conditions. 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 "x""s" with respect to the multiplicative Haar measure,
formula_5, which is invariant
under dilation formula_6, so that
formula_7 the two-sided Laplace transform integrates with respect to the additive Haar measure formula_8, which is translation invariant, so that formula_9.
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.
Examples.
Cahen–Mellin integral.
The Mellin transform of the function formula_12 is
where formula_14 is the gamma function. formula_14 is a meromorphic function with simple poles at formula_16. Therefore, formula_14 is analytic for formula_18. Thus, letting formula_19 and formula_20 on the principal branch, the inverse transform gives
This integral is known as the Cahen–Mellin integral.
Polynomial Functions.
Since formula_22 is not convergent for any value of formula_23, the Mellin transform is not defined for polynomial functions defined on the whole positive real axis. However, by defining it to be zero on different sections of the real axis, it is possible to take the Mellin transform. For example, if
then
Thus formula_26 has a simple pole at formula_27 and is thus defined for formula_28. Similarly, if
then
Thus formula_26 has a simple pole at formula_32 and is thus defined for formula_33.
Exponential Functions.
For formula_34, let formula_35. Then
Zeta Function.
It is possible to use the Mellin transform to produce one of the fundamental formulas for the Riemann zeta function, formula_37. Let formula_38. Then
Thus,
Generalized Gaussian.
For formula_41, let formula_42 (i.e. formula_43 is a generalized Gaussian distribution without the scaling factor.) Then
In particular, setting formula_45 recovers the following form of the gamma function
Fundamental strip.
For formula_47, let the open strip formula_48 be defined to be all formula_49 such that formula_50 with formula_51 The fundamental strip of formula_52 is defined to be the largest open strip on which it is defined. For example, for formula_53 the fundamental strip of
is formula_55 As seen by this example, the asymptotics of the function as formula_56 define the left endpoint of its fundamental strip, and the asymptotics of the function as formula_57 define its right endpoint. To summarize using Big O notation, if formula_43 is formula_59 as formula_56 and formula_61 as formula_62 then formula_52 is defined in the strip formula_55
An application of this can be seen in the gamma function, formula_65 Since formula_66 is formula_67 as formula_56 and formula_69 for all formula_70 then formula_71 should be defined in the strip formula_72 which confirms that formula_14 is analytic for formula_74
As an isometry on "L"2 spaces.
In the study of Hilbert spaces, the Mellin transform is often posed in a slightly different way. For functions in formula_75 (see Lp space) the fundamental strip always includes formula_76, so we may define a linear operator formula_77 as
In other words, we have set
This operator is usually denoted by just plain formula_81 and called the "Mellin transform", but formula_77 is used here to distinguish from the definition used elsewhere in this article. The Mellin inversion theorem then shows that formula_77 is invertible with inverse
Furthermore, this operator is an isometry, that is to say formula_86 for all formula_87 (this explains why the factor of formula_88 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 "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 formula_90 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":
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 particularize 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 formula_105.
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.