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Application of the Generalized Shift Operator to the Hankel Transform Natalie Baddour

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Application of the Generalized Shift Operator to the Hankel Transform Natalie Baddour View metadata, citation and similar papers at core.ac.uk brought to you by CORE provided by Crossref Baddour SpringerPlus 2014, 3:246 http://www.springerplus.com/content/3/1/246 a SpringerOpen Journal SHORT REPORT Open Access Application of the generalized shift operator to the Hankel transform Natalie Baddour Abstract 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. Introduction Background It is well known that the Hankel transform does not satisfy Several definitions of the Hankel transform appear in the a convolution or shift theorem in the simple way as that literature. In this paper, we use the definition of the nth the Fourier and Laplace transforms (Piessens 2000), re- order Hankel transform as defined by Piessens in (Piessens ducing its apparent utility. This follows because the 2000) to define the Hankel transform as Bessel functions do not possess a simple addition formula in the same way that the exponentials satisfy ei(x + y) = eixeiy. Z∞ The operation of convolution between two functions FðÞ¼ρ ℍnðÞ¼frðÞ frðÞJ nðÞρr rdr; ð1Þ consists of a shift in one of the functions, a multiplica- 0 tion with the other function, followed by an integration (or summation for discrete transforms) over all allow- where J (z)isthenth order Bessel function. Here, n may able shifts. Thus, the lack of a convolution theorem for n be an arbitrary real or complex number. However, an inte- the Hankel transform follows because of the lack of a gral transform needs to be invertible in order to be useful simple expression for the shift of a function in the and this restricts the allowable values of n.Ifn is real and Hankel transform domain. n > − 1/2, and under suitable conditions of integrability of Levitan introduced the idea of a generalized displace- the function, the transform is self-reciprocating and the in- ment operator (Levitan 2002). As useful as this concept version formula is given by might be, to the best of the author’s knowledge it does not appear to have seen much application in the physics or en- Z∞ gineering literature. In this paper, Levitan’s generalized dis- ðÞ¼ ðÞρ ðÞρ ρ ρ ð Þ placement operator is made concrete via application to fr F J n r d 2 the Hankel transform. We show that this leads to shift 0 and convolution rules for the Hankel transform. By way of several examples, the Hankel (generalized) shift is com- Thus, the Hankel transforms takes a function f(r)in pared to the standard simple shift. 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). Correspondence: [email protected] Department of Mechanical Engineering, University of Ottawa, Ottawa, Ontario K1N 6N5, Canada © 2014 Baddour; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited. Baddour SpringerPlus 2014, 3:246 Page 2 of 6 http://www.springerplus.com/content/3/1/246 Generalized shift for the Hankel Transform where the order of integration has been reversed (Levitan 2002) introduced the idea of a generalized shift and where: operator. Using Levitan’s definition, the generalized shift Z∞ operator Rr0 indicates a shift of r acting on the function 0 ðÞ¼; ; ðÞρ ðÞρ ðÞρ ρ ρ ð Þ f(r) is defined by the formula (Levitan 2002). Dn r r0 x J n r J n r0 J n x d 6 0 Z∞ r0 R frðÞ¼frðÞ¼jr0 FðÞρ J nðÞρr0 J nðÞρr ρdρ ð3Þ The interpretation of the generalized shift operator from Equation (5) allows the shift to be seen directly 0 as an operation on the original untransformed func- where F(ρ) is the Hankel transform of f(r) and we write tion. The definition as given in (3) allows for better r0 the generalized-shifted function as frrð j 0Þ ≡ R frðÞ,asa physical interpretation of the definition (in particu- reminder that the shifted function is now a function of lar in comparison with the familiar Fourier trans- r0 forms) and also permits the simple proofs to follow r and r0. The operator R acting on the function f(r) indicates a shift in f(r)byr0. The intuitive definition of for the shift and convolution rules. the Hankel shift (generalized shift), as defined in (3), is that of the inverse Hankel transform of F(ρ)Jn(ρr0), whereas the unshifted function would be the inverse Generalized shift rule Hankel transform of F(ρ) alone, without the multipli- In keeping with the standard shift-modulation rule of the Fourier transform, the definition of the Hankel cation by Jn(ρr0). In essence, Equation (3) says that multi- plication by Jn(ρr0) in the Hankel domain is equivalent to generalized shift allows for a similar rule to be derived a generalized shift in the spatial domain. for the Hankel transform. The Hankel transform of This definition follows the same expression used for the generalized-shifted function is given by application the Fourier transform, that is: of Equation (1) to the generalized-shifted function in (3) as Z∞ Z∞ ÀÁ 8 9 1 − ω ω 1 ω à ω Z∞ Z∞ FðÞω e i t0 ei tdω ¼ FðÞω ei t0 ei tdω < = 2π 2π −∞ −∞ ℍnðÞ¼frðÞjr0 FxðÞJ nðÞxr0 J nðÞxr xdx J nðÞρr rdr Z∞ : ; 1 ωðÞ− 0 0 ¼ FðÞω ei t t0 dω ¼ ftðÞ−t Z∞ Z∞ 2π 0 −∞ ¼ FxðÞJ nðÞxr0 J nðÞxr J nðÞρr rdr xdx ð4Þ 0 0 ð7Þ In (4), the star denotes the complex conjugate and fol- lows the definition of generalized shift as given by Levi- where the order of integration has been reversed. tan. As previously pointed out, the simple shift, f(t − t0), Using the orthogonality (closure) of the Bessel func- ω − ω ωðÞ− follows from the definition because ei te i t0 ¼ ei t t0 . tions (Watson 1995; Abramowitz & Stegun 1964) For the Hankel transform with Bessel functions, no sim- Z∞ ple equivalent expression exists, but the general struc- 1 J ðÞxr J ðÞρr rdr ¼ δðÞx−ρ ; ð8Þ ture of the shift operation for the Fourier transform n n x (left-hand side of Equation (4)) is the same. 0 It is noted that using the definition of Hankel trans- r0 it then follows that: form given in (1), the shifted function R frðÞ¼frrð j 0Þ can also be written as Z∞ δðÞx−ρ 8 9 ℍnðÞ¼frðÞjr0 FxðÞJ nðÞxr0 xdx Z∞<Z∞ = x r0 0 R frðÞ¼frðÞ¼jr0 fxðÞJ nðÞρx xdx J nðÞρr0 J nðÞρr ρdρ : ; ¼ FðÞρ J nðÞρr0 ð9Þ 0 0 Z∞ Z∞ ¼ fxðÞ J nðÞρr0 J nðÞρr J nðÞρx ρdρxdx In other words, a (generalized) shift in the spatial do- main is equivalent to multiplication by Jn(ρr0)inthe Z0∞ 0 Hankel domain or, if f(r) ⇔ F(ρ)thenf(r|r0) ⇔ F(ρ)Jn ¼ fxðÞDnðÞr; r0; x xdx (ρr0). This follows the same rule as for the Fourier 0 transform where the Fourier transform of f(r − r0)is ð Þ − ω 5 given by FðÞω e i r0 . Baddour SpringerPlus 2014, 3:246 Page 3 of 6 http://www.springerplus.com/content/3/1/246 Modulation rule Z∞Z∞ ℍ ðÞ¼ðÞà ðÞ ðÞðÞj ðÞρ Since the Hankel transform is self-reciprocal, a “modu- n g H f r gr0 frr0 r0dr0J n r rdr 0 0 8 9 lation” rule similar to that of the Fourier transform can Z∞Z∞ <Z∞ = ρ ¼ ðÞ ðÞ ðÞðÞ ðÞρ easily be derived. The Hankel transform of Jn(r 0)f(r)is gr0 : FxJ n xr0 J n xr xdx; r0dr0J n r rdr given by 0 0 0 ð13Þ Z∞ ÀÁÀÁ ÀÁ ℍ ρ ðÞ ¼ ðÞ ρ ðÞρ Interchanging the order of integration gives: it then n J n r 0 fr frJ n r 0 J n r rdr ð Þ 10 follows that 0 ÀÁ ρ ¼ ρjρ ¼ 0 ðÞρ F 0 R F Z∞Z∞Z∞ ℍnðÞ¼ðÞgÃH f ðÞr grðÞ0 J nðÞxr0 r0dr0 JnðÞxr JnðÞρr rdrFðÞ x xdx where the definition of the shifted transform follows 0 0 |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}0 by replacing r, r0, ρ, f with ρ, ρ0, r, F in the definition ¼GxðÞ Z∞ Z∞ (3)sothat ¼ GxðÞFxðÞ J nðÞxr J nðÞρr rdr xdx ¼ GðÞρ FðÞρ 0 |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}0 Z∞ 1 ÀÁ ÀÁ ¼ δðÞx−ρ ρ x R 0 FðÞ¼ρ F ρjρ ¼ frðÞJ n rρ J nðÞρr rdr ð11Þ 0 0 ð Þ 0 14 In the preceding Equation, (14), the definition of the ⇔ ρ ÀÁIn other words,À if f(r) F( ) then it follows that J n Hankel transform of g(r) has been used, in addition to ρ ρ ðÞ⇔ ρρ Þ¼ 0 ðÞρ r 0 fr F 0 R F . the orthogonality of the Bessel functions. Equation (14) clearly states that the Hankel transform of the Hankel convolution is the product of the Hankel transforms, Convolution rule again in parallel with the standard result of Fourier With the definition of the generalized shift, we define transforms.
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