Generalization of Fourier and Mellin transforms

Integral transform, such as Fourier and Mellin are helpful in providing techniques for solving problems in differential equations. It can be used as a mathematical tool to solve these problems. In this paper, we find the correlation of Fourier transform and Mellin transforms of some function f:R→C and characterizing condition in terms of Mellin transform. This will not only be helpful in solving various differential and integral equations, but also have versatile applications in many fields of applied science.

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Introduction

The Fourier and Mellin transforms have been used collectively in many fields separately. On combining these two transforms i.e. Fourier-Finite Mellin transforms is obtained. Fourierfinite Mellin transforms are used for solving differential and integral equations. The algebraic properties of Fourier and Mellin transforms are (briefly) worked out in a series of exercises in [8] . For the more algebraically inclined, one can develop an abstract theory of convolution and Fourier analysis on groups. See [7] , or [14] for a full treatment [1, 19 , 11] , and [15] . all cover the Mellin transform, the last two in the probability context [3] . contains an extensive table of Mellin transforms (as well as Fourier, Laplace, and other transforms). A more abstract view is provided by [20] , which includes a treatment of integral transforms of (Schwartz) distributions [5] . contains a very complete treatment of properties of the Mellin transform, with proofs. But here in this paper we find the correlation of Fourier and Infinite Mellin transforms and how Mellin transform can be obtained from a Fourier transform for some function : → , which is helpful for solving differential equations with manifold applications. Important and interesting as those cited references are, we know that the Fourier transform has an inherent complex structure and is (more) naturally defined on R (rather than R+).Here we consider [Q] and use the below criteria (a) The integral transform is unitary. (b) The integral kernel is Fourier. (a) And (b) are understood by well-known Fourier-transform properties:

so that we have = , where denotes the identity operator, and the kernel is ( ) = √2 Thus, the Fourier transform itself is unitary, with a kernel ( ), satisfying both (a)

A characterizing condition on ( ) in terms of the Mellin transform has been obtained in [6] , [15] ( ) (1 -) = 1

is the Mellin transform of .

Interrelation of Mellin and Fourier transform

The Mellin transform of with argument ∈ is defined as

where ≤ ( ) ≤ (if 1 converges for = ∈ , then it can be easily shown that it converges for = + , ∈ ) For ( ) = , the inverse Mellin transform is

For the existence of inverse the condition is that ( ) is analytic in a strip ( , ) × (-∞, ∞) such that ∈ ( , ) [5]. In many engineering texts, statistics, the Fourier transform is defined as

Using the transform and substituting = -( ) for all real ≥ 0

(An aside on the substitution = -( ). The factor of 2π is a consequence of the way we define the Fourier transform. In statistics, and in many engineering texts, the Fourier transform is defined as ( ) = ( ) (essentially measuring frequency in radians per time unit instead of cycles per time unit), which simplifies the derivation of the Mellin transform from the Fourier transform. For a summary of the different ways the Fourier transform and its inverse are represented, see [13] .) The same technique can be used to derive Mellin inverse formula from Fourier inverse transform as

In some cases the transformation provides an easier way to invert Mellin transforms, by using Fourier inversion techniques.

Special case

For a suitable function : → , The Fourier transform of is the integral

for any suitable σ Thus, the Mellin inversion formula says that if the function and are well enough behaved then = iff = Now, easily the integral given below can also be evaluated

Generalized Characterizing condition in terms of Mellin transform Theorem: Let ( ) be a Fourier kernel (on R) such that (a) and (b) are satisfied. This holds if and only if

where ҡ and ҡ are respectively the Mellin transform of and of the Fourier kernel , ҡ ± ( ) = ҡ ± ( ̅ ) , is 2×2 identity matrix and can be analytically continued on .

Proof: For any function with compact support on , we have ( ) + ( ) let ( ) be the Fourier kernel satisfying (a) and (b), and let

taking Mellin transform of (1.2) and (1.3) and using

where ± , ҡ ± and ± are respectively the Mellin transform of ± , ± , ± , we thus have

similarly for reciprocal relation

from (1.4) and (1.5) we conclude (1.1).

Further reading:

Mathematical science is not the only one application area for the Mellin transform and Fourier transform, but many other applications also make the Fourier transform and its variants universal elsewhere in almost all branches of science and engineering, i.e, it has versatile applications in many fields. The Mellin transform is used in computer science for analysis of algorithms (see, for example, [17, ch. 9-10] ); it has applications to analytic number theory [10] ; and Mellin himself developed it in connection with his researches in the theory of functions, number theory, and partial differential equations [18] . The result (1.1) can also be extended in 3 × 3 identity matrix, that will further improve the characterizing condition of Mellin transform.

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