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 is named after the Finnish mathematician Hjalmar Mellin, who introduced it in a paper published 1897 in Acta Societatis Scientiarum Fennicæ.[1]
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; in terms of the Mellin transform and of the two-sided Laplace transform defined above
This integral is known as the Cahen–Mellin integral.[3]
Polynomial functionsedit
Since is not convergent for any value of , 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 has a simple pole at and is thus defined for . Similarly, if
then
Thus has a simple pole at and is thus defined for .
Exponential functionsedit
For , let . Then
Zeta functionedit
It is possible to use the Mellin transform to produce one of the fundamental formulas for the Riemann zeta function, . Let . Then
In particular, setting recovers the following form of the gamma function
Power series and Dirichlet seriesedit
Generally, assuming necessary convergence, we can connect Dirichlet series and related power series
by the formal identity involving Mellin transform:[4]
Fundamental stripedit
For , let the open strip be defined to be all such that with The fundamental strip of is defined to be the largest open strip on which it is defined. For example, for the fundamental strip of
is As seen by this example, the asymptotics of the function as define the left endpoint of its fundamental strip, and the asymptotics of the function as define its right endpoint. To summarize using Big O notation, if is as and as then is defined in the strip [5]
An application of this can be seen in the gamma function, Since is as and for all then should be defined in the strip which confirms that is analytic for
The domain shift is conditional and requires evaluation against specific convergence behavior.
Valid only if the integral exists.
Valid only if the integral exists.
Multiplicative convolution
Multiplicative convolution (generalized)
Multiplicative convolution (generalized)
Multiplication. Only valid if integral exists. See Parseval's theorem below for conditions which ensure the existence of the integral.
Parseval's theorem and Plancherel's theoremedit
Let and be functions with well-defined
Mellin transforms
in the fundamental strips .
Let with .
If the functions and
are also square-integrable over the interval , then Parseval's formula holds:
[6]
The integration on the right hand side is done along the vertical line that
lies entirely within the overlap of the (suitable transformed) fundamental strips.
We can replace by . This gives following alternative form of the theorem:
Let and be functions with well-defined
Mellin transforms
in the fundamental strips .
Let with and
choose with .
If the functions and
are also square-integrable over the interval , then we have
[6]
We can replace by .
This gives following theorem:
Let be a function with well-defined Mellin transform in the fundamental strip .
Let with .
If the function is also square-integrable over the interval , then Plancherel's theorem holds:[7]
As an isometry on L2 spacesedit
In the study of Hilbert spaces, 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).
In probability theoryedit
In probability theory, the Mellin transform is an essential tool in studying the distributions of products of random variables.[8] 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[9]
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.[9]
The Mellin transform of a random variable X uniquely determines its distribution function FX.[9] 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 product is equal to the product of the Mellin transforms of X and Y:[10]
Problems with Laplacian in cylindrical coordinate systemedit
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 treated with the Mellin transform,[11] since:
For example, the 2-D Laplace equation in polar coordinates is the PDE in two variables:
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 .
Applicationsedit
The Mellin Transform is widely used in computer science for the analysis of algorithms[12] 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.
^Hardy, G. H.; Littlewood, J. E. (1916). "Contributions to the Theory of the Riemann Zeta-Function and the Theory of the Distribution of Primes". Acta Mathematica. 41 (1): 119–196. doi:10.1007/BF02422942. (See notes therein for further references to Cahen's and Mellin's work, including Cahen's thesis.)
^Bhimsen, Shivamoggi, Chapter 6: The Mellin Transform, par. 4.3: Distribution of a Potential in a Wedge, pp. 267–8
^Philippe Flajolet and Robert Sedgewick. The Average Case Analysis of Algorithms: Mellin Transform Asymptotics. Research Report 2956. 93 pages. Institut National de Recherche en Informatique et en Automatique (INRIA), 1996.
^A. Liam Fitzpatrick, Jared Kaplan, Joao Penedones, Suvrat Raju, Balt C. van Rees. "A Natural Language for AdS/CFT Correlators".
^A. Liam Fitzpatrick, Jared Kaplan. "Unitarity and the Holographic S-Matrix"
^A. Liam Fitzpatrick. "AdS/CFT and the Holographic S-Matrix", video lecture.
^Jacqueline Bertrand, Pierre Bertrand, Jean-Philippe Ovarlez. The Mellin Transform. The Transforms and Applications Handbook, 1995, 978-1420066524. ffhal-03152634f
Referencesedit
Lokenath Debnath; Dambaru Bhatta (19 April 2016). Integral Transforms and Their Applications. CRC Press. ISBN 978-1-4200-1091-6.
Galambos, Janos; Simonelli, Italo (2004). Products of random variables: applications to problems of physics and to arithmetical functions. Marcel Dekker, Inc. ISBN 0-8247-5402-6.
Paris, R. B.; Kaminski, D. (2001). Asymptotics and Mellin-Barnes Integrals. Cambridge University Press. ISBN 9780521790017.
Polyanin, A. D.; Manzhirov, A. V. (1998). Handbook of Integral Equations. Boca Raton: CRC Press. ISBN 0-8493-2876-4.
Bracewell, Ronald N. (2000). The Fourier Transform and Its Applications (3rd ed.).
Erdélyi, Arthur (1954). Tables of Integral Transforms. Vol. 1. McGraw-Hill.
Titchmarsh, E.C. (1948). Introduction to the Theory of Fourier Integrals (2nd ed.).