Application of the generalized shift operator to the Hankel transform
© Baddour; licensee Springer. 2014
Received: 2 March 2014
Accepted: 9 April 2014
Published: 14 May 2014
It is well known that the Hankel transform possesses neither a shift-modulation nor a convolution-multiplication rule, both of which have found many uses when used with other integral transforms. In this paper, the generalized shift operator, as defined by Levitan, is applied to the Hankel transform. It is shown that under this generalized definition of shift, both convolution and shift theorems now apply to the Hankel transform. The operation of a generalized shift is compared to that of a simple shift via example.
It is well known that the Hankel transform does not satisfy a convolution or shift theorem in the simple way as that the Fourier and Laplace transforms (Piessens 2000), reducing its apparent utility. This follows because the Bessel functions do not possess a simple addition formula in the same way that the exponentials satisfy ei(x + y) = e ix e iy . The operation of convolution between two functions consists of a shift in one of the functions, a multiplication with the other function, followed by an integration (or summation for discrete transforms) over all allowable shifts. Thus, the lack of a convolution theorem for the Hankel transform follows because of the lack of a simple expression for the shift of a function in the Hankel transform domain.
Levitan introduced the idea of a generalized displacement operator (Levitan 2002). As useful as this concept might be, to the best of the author’s knowledge it does not appear to have seen much application in the physics or engineering literature. In this paper, Levitan’s generalized displacement operator is made concrete via application to the Hankel transform. We show that this leads to shift and convolution rules for the Hankel transform. By way of several examples, the Hankel (generalized) shift is compared to the standard simple shift.
Thus, the Hankel transforms takes a function f(r) in the spatial r domain and transforms it to a function F(ρ) in the frequency ρ domain. This relationship is denoted symbolically as f(r) ⇔ F(ρ).
Generalized shift for the Hankel Transform
where F(ρ) is the Hankel transform of f(r) and we write the generalized-shifted function as , as a reminder that the shifted function is now a function of r and r0. The operator acting on the function f(r) indicates a shift in f(r) by r0. The intuitive definition of the Hankel shift (generalized shift), as defined in (3), is that of the inverse Hankel transform of F(ρ)J n (ρr0), whereas the unshifted function would be the inverse Hankel transform of F(ρ) alone, without the multiplication by J n (ρr0). In essence, Equation (3) says that multiplication by J n (ρr0) in the Hankel domain is equivalent to a generalized shift in the spatial domain.
In (4), the star denotes the complex conjugate and follows the definition of generalized shift as given by Levitan. As previously pointed out, the simple shift, f(t − t0), follows from the definition because . For the Hankel transform with Bessel functions, no simple equivalent expression exists, but the general structure of the shift operation for the Fourier transform (left-hand side of Equation (4)) is the same.
The interpretation of the generalized shift operator from Equation (5) allows the shift to be seen directly as an operation on the original untransformed function. The definition as given in (3) allows for better physical interpretation of the definition (in particular in comparison with the familiar Fourier transforms) and also permits the simple proofs to follow for the shift and convolution rules.
Generalized shift rule
In other words, a (generalized) shift in the spatial domain is equivalent to multiplication by J n (ρr0) in the Hankel domain or, if f(r) ⇔ F(ρ) then f(r|r0) ⇔ F(ρ)J n (ρr0). This follows the same rule as for the Fourier transform where the Fourier transform of f(r − r0) is given by .
In other words, if f(r) ⇔ F(ρ) then it follows that .
Equation (12) is in keeping with the typical definition of a convolution in the radial domain, where the simple shift f(r − r0) has been replaced with the generalized shift f(r|r0). Furthermore, we use the notation * H to denote that this is a Hankel convolution, meaning that the generalized Hankel shift operator is used instead of the simple shift operator. It is noted that other authors define a Hankel convolution without reference to the generalized shift operator. In all those cases, the integral of a triple product of Bessel functions is used to define the Hankel convolution, for example in (Tuan & Saigo 1995; Malgonde & Gaikawad 2001; de Sousa Pinto 1985; Belhadj & Betancor 2002). The mathematical properties of Hankel convolutions are analyzed in (Tuan & Saigo 1995; Malgonde & Gaikawad 2001; de Sousa Pinto 1985; Belhadj & Betancor 2002; Betancor & Marrero 1993; Betancor & Marrero 1995; Cholewinski & Haimo 1966).
The primary utility of the generalized shift function would appear to be that it is the function that permits the standard shift, modulation, multiplication and convolution rules to apply when using the Hankel transform.
Generally speaking, the Hankel transform alone (without an accompanying angular coordinate Fourier transform to turn it into a two- dimensional Fourier transform) is most used in physical systems that have radial symmetry. Once the system is shifted in the radial direction - as is necessary to take a convolution - radial symmetry is lost and thus the proper, physically-meaningful transform would be a full 2D Fourier transform in polar coordinates. We showed in (Baddour 2009; Baddour 2011) that if a full (radial and angular) shift is taken in defining the convolution, then the 2D Fourier transform in polar coordinates does possess the standard shift, modulation, multiplication and convolution rules.
What we have demonstrated in this paper is that if it is desirable to use only the Hankel transform and work with only the radial coordinate, then the Hankel transform does possess the standard shift, modulation, multiplication and convolution rules – but only when the generalized definition of the shift is employed, not with the simple shift.
We have shown that the Hankel transform does possess the standard shift, modulation, multiplication and convolution rules – but only when the generalized version of the shift is employed, not with the simple shift. We demonstrated by way of a simple example that the generalized shift and simple shift are not the same and thus not interchangeable for simulations of physical systems. The value of the generalized shift is that it permits the standard Fourier rules to apply to the Hankel transform. For the purposes of calculating physically meaningful convolutions, the simple shift in the radial coordinate should also be accompanied with an angular shift. A physically meaningful convolution implies integration with a shift over all physical coordinates allowed by the geometry of the problem. Fortunately, the full 2D Fourier transform in polar coordinates possesses the desired shift and convolution rules.
This research was financially supported by the National Science and Engineering Research Council of Canada.
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