Bargmann's versus of the quaternionic fractional Hankel transform
BBARGMANN’S VERSUS OF THE QUATERNIONIC FRACTIONAL HANKELTRANSFORM
ABDELATIF ELKACHKOURI, ALLAL GHANMI, AND ALI HAFOUDA
BSTRACT . We investigate the quaternionic extension of the fractional Fourier transformon the real half-line leading to fractional Hankel transform. This will be handled `a laBargmann by means of hyperholomorphic second Bargmann transform for the slice Bergmanspace of second kind. Basic properties are derived including inversion formula and Plancherelidentity.
1. I
NTRODUCTION
The fractional Fourier transform (FrFT), which is special generalizationof the Fourier integral transform, is a powerful tool in many fields of re-search including mathematics, physics and engineering sciences [1, 15,12]. Its introduction goes back to 1929 when was considered implicitly inWiener ’s work [18], discussing the extension of certain results of H. Weyland leading later to Fourier developments of fractional order. Mainly,Wiener sets out to find a one–parameter family of unitary integral opera-tors F α ϕ ( x ) : = (cid:90) + ∞ − ∞ K α ( x , v ) ϕ ( v ) dv on L ( R ) , for which the n –th Hermite function h n ( x ) = H n ( x ) e − x /2 is aeigenfunction with e in α as corresponding eigenvalue. The explicit Wienerformula for the kernel function K α is a limiting case of Mehler ’s formulafor the Hermite functions. This fact was rediscovered sixty years later inquantum mechanics by Namias [13], and showed earlier by H ¨ormander[10].Another and elegant way to define the standard FrFT is due to Bargmann[3] (see also [4, 9, 17]) and lies on the classical Segal–Bargmann transform B . Indeed, the transform R θ : = B − ◦ T θ ◦ B , with Γ θ f ( z ) : = f ( θ z ) , defines Mathematics Subject Classification.
Primary 30G35.
Key words and phrases.
Fractional Fourier transform; Fractional Hankel transform; Slice hyperholomor-phic Bergman space; Second Bargmann transform; Laguerre polynomials; Bessel functions. a r X i v : . [ m a t h . C V ] M a r ABDELATIF ELKACHKOURI, ALLAL GHANMI, AND ALI HAFOUD a unitary homeomorphism transform on L ( R d ) and satisfies R θ h n ( x ) = e in α h n ( x ) .Apparently, except Bargmann’s work [3] for FrFT on L ( R d ) in connec-tion with the Fock–Bargmann space, there is no general abstract approachin the literature for constructing fractional transform associated to giveninvertible integral transform S X , Y : H X −→ H Y on an arbitrary infiniteseparable functional Hilbert space H X = L ( X ; ω X d λ ) . In absence of suchgeneral formalism, the authors in [5] have recently tried to extend, in apartial way, the Fourier fractional formalism to the generalized Laguerrefunctions by means of their relation to Hermite polynomials.In the present paper, we provide such abstract formalism `a la Bargmann.Namely, we deal with integral transforms of the form S − X , Y ◦ T θ ◦ S X , Y , as-sociated to given special invertible integral transform S X , Y ϕ ( y ) = (cid:90) X R ( x , y ) ϕ ( x ) ω X ( x ) d λ ( x ) and given appropriate action of a group G . We show that the performedfractional integral transform inherits numerous properties from the onesof S X , Y . The explicit computation shows that the kernel function of this in-tegral transform can be expressed explicitly in terms of the kernel function R ( x , y ) . As concrete application, we deal with a special quaternionic frac-tional Fourier transform (QFrFT) acting on the right quaternionic Hilbertspace L α H ( R + ) = L H (cid:0) R + , x α e − x dx (cid:1) , α > L αθ ϕ ( y ) : = (cid:90) ∞ K αθ ( x , y ) ϕ ( x ) dx , (1.1)whose kernel function can be shown to be given in terms of the mod-ified Bessel function I α , verifying L αθ ( ϕ α n ( x )) = θ n ϕ α n ( x ) and leading inparticular to a variational definition of the well-known fractional Han-kel transform [14, 11]. We also prove that L αθ is continuous, interpolates continuously the identity operator to the Fourier-Bessel (Hankel) trans-form and satisfies the index law (semi-group property) L αθ ◦ L θη = L αθψ , sothat the inversion formula reads simply ( L αθ ) − = L αθ − . The consideredfamily of QFrFT for L α H ( R + ) appears embedded in a strongly continuousone-parameter group of unitary operators when | θ | =
1. The expositionof these ideas in the quaternionic setting add some technical difficultieswhich we overcome using tools from the theory of slice regular functions.The paper is organized as follows. In Section 2, we present an abstractformalism for constructing quaternionic fractional integral transforms bymeans of eigenvalue equation involving orthogonal basis of certain quater-nionic Hilbert space. Section 3 is devoted to the reconstruction of quater-nionic fractional Hankel transform for L α H ( R + ) by Bargmann versus, andshow how to derive in a simple way their basic properties such as thePlancherel and inversion formulas.2. P RELIMINARIES
In this section, we review the two classical ways of constructing FrFTthat we extend in a natural way to any arbitrary infinite functional leftquaternionic separable Hilbert space H X on given set X . We provide then`a la Namias a concrete example giving rise to the fractional Hankel trans-form for the quaternionic left Hilbert space L α H ( R + ) .2.1. Abstract formalism for fractional integral transform.
Let H X and H Y be two arbitrary infinite functional left quaternionic separable Hilbertspaces with orthonoromal bases ϕ n and ψ n on X and Y , respectively. Thecorresponding inner scalar products are given by (cid:104) ϕ , φ (cid:105) H X = (cid:90) X ϕ ( x ) φ ( x ) ω X ( x ) dx and (cid:104) Ψ , Φ (cid:105) H Y = (cid:90) Y Ψ ( y ) Φ ( y ) ω Y ( y ) dy ,respectively, for some weight functions ω X and ω Y . Associated to thedata ( X , H X , ϕ n ) and ( Y , H Y , ψ n ) , we consider the integral transform T XY : ABDELATIF ELKACHKOURI, ALLAL GHANMI, AND ALI HAFOUD H X −→ H Y of the form T XY ( ϕ )( y ) = (cid:90) X R ( x , y ) ϕ ( x ) ω X ( x ) dx .We assume that T XY is well defined on H X such that T XY ( ϕ n ) = ψ n . This isequivalent to say that kernel function R ( x , y ) on X × Y can be expandedas R ( x , y ) = ∞ ∑ n = ϕ n ( x ) ψ n ( y ) whenever the series in the right-hand side is uniformly and absolutelyconvergent. Subsequently, T XY is an invertible integral kernel transform,whose inverse is given by T − XY ψ ( x ) = (cid:90) Y R ( x , y ) ψ ( y ) ω Y ( y ) dy for x ∈ X and ψ ∈ H Y . Now, we need to special action of some group G on Y that we will extend to H Y by considering Γ : G × H Y −→ H Y ; Γ ( g , ψ ) = Γ g ( ψ ) , so that the corresponding diagrams H X T XY (cid:47) (cid:47) F g (cid:15) (cid:15) (cid:9) H Y Γ g (cid:15) (cid:15) (cid:9) G × H Y Γ (cid:111) (cid:111) Γ g (cid:15) (cid:15) H X H YT − XY (cid:111) (cid:111) G × H Y Γ (cid:111) (cid:111) becomes commutative, which means that Γ satisfies Γ ( g , ψ )( y ) = Γ g ( ψ )( y ) = ψ ( Γ g . y ) ; y ∈ Y , ψ ∈ H Y .We then perform the fractional transform F rg = T − XY ◦ Γ g ◦ T XY ; g ∈ G . For every ψ ∈ H Y , we have F rg ( ϕ )( x ) = (cid:90) y ∈ Y R ( x , y ) Γ g ( T XY ( ϕ ))( y ) ω Y ( y ) dy = (cid:90) y ∈ Y R ( x , y ) (cid:18) (cid:90) x (cid:48) ∈ X R ( x (cid:48) , Γ g ( y )) ϕ ( x (cid:48) ) ω X ( x (cid:48) ) dx (cid:48) (cid:19) ω Y ( y ) dy Fubini = (cid:90) x (cid:48) ∈ X (cid:102) R g ( x (cid:48) , x ) ϕ ( x (cid:48) ) ω X ( x (cid:48) ) dx (cid:48) ,where R g ( x (cid:48) , x ) stands for (cid:102) R g ( x (cid:48) , x ) = (cid:10) R ( x (cid:48) , Γ g ( y )) , R ( x , y ) (cid:11) H Y .An expansion of (cid:102) R g ( z , x ) , at least formally, is the following (cid:102) R g ( x (cid:48) , x ) = ∞ ∑ m = ∞ ∑ n = ϕ n ( x ) (cid:104) ψ n , Γ g ψ m (cid:105) H Y ϕ m ( x (cid:48) )= ∞ ∑ n = ϕ n ( x ) χ n ( g ) ϕ n ( x (cid:48) ) = : R g ( x (cid:48) , x ) . (2.1)The last equality follows under the additional assumption that Γ g ψ m ( y ) = ψ m ( Γ g ( y )) = χ m ( g ) ψ m ( y ) .According to the above discussion, we reformulate the following defini-tions. Definition 2.1.
If the series in the right-hand side of (2.1) converges absolutelyand uniformly to R g ( x (cid:48) , x ) , then F rg ( ϕ )( x ) : = (cid:90) x (cid:48) ∈ X R g ( x (cid:48) , x ) ϕ ( x (cid:48) ) ω X ( x (cid:48) ) dx (cid:48) defines a like-fractional Fourier transform for the data ( H X , ϕ n , χ n ) . Definition 2.2.
We call fractional Fourier transform associated to T XY and Γ theintegral transform (cid:102) F rg ( ϕ )( x ) = T − XY Γ g T XY ( ϕ )( x ) = (cid:90) x (cid:48) ∈ X (cid:102) R g ( x (cid:48) , x ) ϕ ( x (cid:48) ) ω X ( x (cid:48) ) dx (cid:48) ABDELATIF ELKACHKOURI, ALLAL GHANMI, AND ALI HAFOUD with (cid:102) R g ( x (cid:48) , x ) = (cid:10) R ( x (cid:48) , Γ g ( y )) , R ( x , y ) (cid:11) H Y . (2.2) Remark 2.3.
We have F rg ( ϕ n ) = ϕ n χ n ( g ) . This gives an integral representa-tion for ϕ n . Quaternionic fractional Hankel transform ( `a la Namias).
We con-sider the right quaternionic Hilbert space L α H ( R + ) ; α >
0, of all quaternionic-valued functions on the half real line R + that are square integrable withrespect to the inner product (cid:104) ϕ , ψ (cid:105) α = (cid:90) R + ϕ ( x ) ψ ( x ) x α e − x dx .We denote by (cid:107)·(cid:107) α the associated norm. A complete orthonormal systemin L α H ( R + ) is given by the functions ϕ n ( x ) = (cid:18) n ! Γ ( α + n + ) (cid:19) L ( α ) n ( x ) , (2.3)where L ( α ) n ( x ) denotes the generalized Laguerre polynomials L ( α ) n ( x ) = x − α e x n ! d n dx n (cid:0) x n + α e − x (cid:1) . (2.4)Accordingly, the series function in Definition 2.1 reduces further to Hille–H+ardy identity [2, (6.2.25) p. 288] R αθ ( x , y ) = + ∞ ∑ n = n ! Γ ( α + n + ) θ n L ( α ) n ( x ) L ( α ) n ( y )= − θ (cid:18) θ xy (cid:19) α /2 exp (cid:18) − θ ( x + y ) − θ (cid:19) I α (cid:32) √ θ − θ √ xy (cid:33) (2.5)valid for | θ [ < α , where I α ( ξ ) denotes the mod-ified Bessel function [2, p.199] I α ( ξ ) = (cid:18) ξ (cid:19) n + α ∞ ∑ n = n ! Γ ( α + n + ) (cid:18) ξ (cid:19) n . Thus, we can rewrite the kernel function K αθ ( x , y ) = x α e − x R αθ ( x , y ) as K αθ ( x , y ) = − θ (cid:18) x θ y (cid:19) α /2 exp (cid:18) − x + θ y − θ (cid:19) I α (cid:32) √ θ − θ √ xy (cid:33) , (2.6)so that the corresponding integral operator is well–defined on L α H ( R + ) by L αθ ( ϕ )( y ) = (cid:90) + ∞ K αθ ( x , y ) ϕ ( x ) dx . (2.7)Such transform is closely connected to the fractional Hankel transform[14, 11]. The Laguerre polynomial ϕ α n ( x ) in (2.3) is (left) eigenfunction of L αθ with θ n as corresponding (right) eigenvalue, L αθ ( ϕ α n ( x )) = ϕ α n ( x ) θ n .This readily follows from the definition of L αθ . Moreover, we assert Proposition 2.4.
For | θ | < , the integral transform L αθ defines a continuousk–contraction from L α H ( R + ) into itself with k ≤ ( − | θ | ) − .Proof. Since ( ϕ α n ) n in (2.3) is a complete orthonormal system in L α H ( R + ) ,we can expand any f ∈ L α H ( R + ) as f ( x ) = + ∞ ∑ n = ϕ α n c n for some c n ∈ H .Hence, using the fact that L αθ ( ϕ α n )( x ) = ϕ α n ( x ) θ n , we get L αθ ( f )( x ) = + ∞ ∑ n = ϕ α n ( x ) θ n c n . (2.8)Using the orthogonality of ϕ α n , we obtain (cid:107)L αθ ( ϕ ) (cid:107) = + ∞ ∑ n = | θ | n | c n | ≤ (cid:32) + ∞ ∑ n = | θ | n (cid:33) (cid:32) + ∞ ∑ n = | c n | n (cid:33) ≤ (cid:18) − | θ | (cid:19) (cid:107) ϕ (cid:107) which requires | θ | < (cid:3) ABDELATIF ELKACHKOURI, ALLAL GHANMI, AND ALI HAFOUD T HE QF R FT FOR L α H ( R + ) In view of the explicit expression of the kernel function in (2.6), we seethat we can consider the limit case of the Hille–Hardy formula which cor-responds to | θ | = θ (cid:54) =
1. We show below that this can recoveredby the formalism presented in Definition 2.2 and specified for L α H ( R + ) ,so that for θ = L α H ( R + ) . To this end, we begin by recalling that the hyper-holomorphic second Bargmann integral transform [7], defined by [ A α slice ϕ ]( q ) = (cid:112) π Γ ( α ) ( − q ) α + (cid:90) + ∞ exp (cid:18) tqq − (cid:19) ϕ ( t ) t α e − t dt , (3.1)is the quaternionic analogue of the complex second Bargmann transformintroduced by Bargmann himself in [3, p.203]. It establishes a unitary iso-metric from L α H ( R + ) onto the slice hyperholomorphic Bergman space (ofsecond kind) on the unit ball B in R , A α slice ( B ) : = S R ( B ) ∩ L α ( B I ) , (3.2)where I ∈ H with I = − B I = B ∩ C I and L α ( B I ) : = (cid:26) f : B −→ H ; (cid:90) B I | f ( z ) | d λ α I ( z ) < + ∞ (cid:27) .Here d λ α I denotes the Bergman measure on the unit disc B I in R given by d λ α I ( z = x + Iy ) = (cid:0) − x − y (cid:1) α − dxdy .Sequentially, we have A α slice ( B ) = (cid:40) f ( q ) = ∞ ∑ n = q n c n ; c n ∈ H , ∞ ∑ n = n ! Γ ( α + n + ) | c n | < + ∞ (cid:41) ,so that the restriction to B I is the classical Bergman space on the unitdisc of C I . It should be mentioned here that the scaler product defining L α ( B I ) , (cid:104) f , g (cid:105) I : = (cid:90) B I f ( z ) g ( z ) d λ α I ( z ) , is independent of I when acting on A α slice ( B ) × A α slice ( B ) , i.e., (cid:104) f , g (cid:105) I = (cid:104) f , g (cid:105) J for any f , g ∈ A α slice ( B ) and any I , J such that I = J = − A α slice is well–definedfrom A α slice ( B ) onto L α H ( R + ) , and is given by [7] [ A α slice ] − f ( t ) = (cid:112) π Γ ( α ) (cid:90) B I exp (cid:18) tzz − (cid:19) ( − | z | ) α − ( − z ) α + f | B I ( z ) dxdy . (3.3)Notice for instance that the definition of A α slice ( B ) is based on the classi-cal one on a given disc B I . This was possible by extending the complexholomorphic functions to the whole B by the representation formula (seefor example [6]). While the transform A α slice in (3.1) is associated to thekernel function A α slice ( x ; q ) : = (cid:112) π Γ ( α ) ( − q ) α + exp (cid:18) xqq − (cid:19) (3.4)on R + × B , and obtained as bilinear generating function involving thefunctions ( ϕ α n ) n in (2.3) and the orthonormal basis of A α slice ( B ) given bythe functions f n ( q ) = (cid:18) Γ ( n + α + ) π Γ ( α ) n ! (cid:19) q n . (3.5)Now, by means of A α slice , its inverse [ A α slice ] − and the angular unitaryoperator Γ θ ( f )( q ) = f ( q θ ) , we perform the transform (cid:102) L αθ : = [ A α slice ] − Γ θ A α slice (3.6)on L α H ( R + ) . Here we consider the Γ θ –action Γ θ q = q θ of G = U H ( ) on B , that we extend to the hyperholomorphic Bergman space A α slice ( B ) byconsidering Γ θ ( f )( q ) = f (cid:63) ( q θ ) : = ∞ ∑ n = q n θ n c n (3.7)for given f ( q ) = ∑ ∞ n = q n c n ∈ A α slice ( B ) . The function q (cid:55)−→ f (cid:63) ( q θ ) isin fact the slice regularization of q (cid:55)−→ f ( q θ ) obtained by making use of the left (cid:63) Ls -product for left slice regular functions f ( q ) = ∞ ∑ n = q n a n and g ( q ) = ∞ ∑ n = q n b n on H defined by [8] ( f (cid:63) Ls g )( q ) = ∞ ∑ n = q n (cid:32) n ∑ k = a k b n − k (cid:33) . (3.8)In particular, we have ( f n ) (cid:63) ( q θ ) : = f n ( q ) θ n , (3.9)and therefore we may prove the following. Proposition 3.1.
For θ ∈ H with | θ | ≤ , the transform (cid:102) L αθ in (3.6) definesa continuous integral transform from L α H ( R + ) onto L α H ( R + ) with norm notexceed . For | θ | = , we have (cid:68) (cid:102) L αθ ϕ , (cid:102) L αθ ψ (cid:69) = (cid:104) ϕ , ψ (cid:105) . Proof.
The operator (cid:102) L αθ in (3.6) is well–defined from L α H ( R + ) into itself ifand only if the action Γ θ leaves the space A α slice ( B ) invariant, which is clearfrom the definition of Γ θ given through (3.7). Moreover, using the fact A α slice ϕ α n = f n as well as (3.9), we get (cid:102) L αθ ( ϕ α n ( y )) = [ A α slice ] − ( f n ( · ) θ n ) ( y ) = ϕ α n ( y ) θ n .In addition, under the condition that | θ | =
1, it is clear that Γ θ preservesthe scalar product in A α slice ( B ) . Indeed, for every f = ∑ ∞ n = f n c n and g = ∑ ∞ n = f n d n ∈ A α slice ( B ) , we have (cid:104) Γ θ f , Γ θ g (cid:105) A α slice ( B ) = ∞ ∑ n , m = c n θ n (cid:104) f n , f m (cid:105) A α slice ( B ) θ m d m = ∞ ∑ n = c n | θ n | d n = (cid:104) f , g (cid:105) A α slice ( B ) . Accordingly, the identity (cid:68) (cid:102) L αθ ϕ , (cid:102) L αθ ψ (cid:69) = (cid:104) ϕ , ψ (cid:105) follows as composition ofoperators preserving scaler product. (cid:3) Corollary 3.2. If | θ | = , then the QFrFT (cid:102) L αθ defines a unitary transform fromL α H ( R + ) into L α H ( R + ) . Remark 3.3.
The family of one–parameter transforms L αθ verifies the semi-groupproperty L αθ ◦ L αη = L αθη , so that its inverse is L θ when θ (cid:54) = . But we do nothave L αθ ◦ L αη = L αη ◦ L αθ in general, for lack of commutativity in H . However, L αθ ◦ L αη = L αθη = L αη ◦ L αθ holds only when θ and ψ belongs to the same slice C I : = R + I R ⊂ H ; I = − . The next result gives the explicit expression of the inverse of (cid:102) L αθ . Proposition 3.4.
For any quaternionic θ (cid:54) = , the inverse of (cid:102) L αθ is given by ( (cid:102) L αθ ) − = A − Γ − θ A = (cid:102) L θ . Proof.
It is immediate form the definition of (cid:102) L αθ and the fact that Γ θ ◦ Γ η = Γ θη . (cid:3) The following result identifies the kernel function given by (2.2), (cid:102) R αθ ( x , y ) : = (cid:104) A α slice ( x ; Γ θ · ) , A α slice ( y ; · ) (cid:105) L α ( B I ) (3.10)of the QFrFT transform [ (cid:102) L αθ ( ϕ )]( y ) = (cid:68) (cid:102) R αθ , ϕ (cid:69) L α H ( R + ) . Theorem 3.5.
The kernel function (cid:102) R αθ ( x , y ) is a left slice regular and coincideswith the kernel function of the fractional Hankel transform on the quaternionicunit ball. Moreover, the explicit expression of (cid:102) L αθ is given by (cid:102) L αθ ϕ ( y ) = e θ y θ − ( − θ )( θ y ) α /2 (cid:90) ∞ x α /2 I α (cid:32) √ θ ( − θ ) √ xy (cid:33) e x θ − ϕ ( x ) dx (3.11) for any θ ∈ H with | θ | ≤ and θ (cid:54) = . Proof.
Notice first that for θ = (cid:102) L αθ reduces further to the identity operator of the Hilbertspace L α H ( R + ) and the R αθ ( x , y ) in (3.10) is to considered as the Diracdelta function. To identify the closed expression of the kernel (cid:102) R αθ ( x , y ) ,we should notice that the Γ θ -action reads Γ θ ( q (cid:55)−→ A α slice ( x ; q )) = ( − q θ ) − α − (cid:63) exp (cid:63) (cid:0) xq θ , [ q θ − ] − (cid:1) ,where exp (cid:63) ( f ( q ) , g ( q )) = ∞ ∑ n = f n (cid:63) ( q ) (cid:63) g n (cid:63) ( q ) n ! .For θ being a non real quaternionic number, there exists a unique imagi-nary unit I θ ; I θ = −
1, such that θ ∈ C I θ ∩ S . By means of (3.10) and theindependence of the scaler product (cid:104) f , g (cid:105) I in I when acting on A α slice ( B ) ,we may write (cid:102) R αθ ( x , y ) : = (cid:104) Γ θ A α slice ( x ; · ) , A α slice ( y ; · ) (cid:105) L α ( B I θ ) = π Γ ( α ) (cid:90) B I exp (cid:0) xz θ z θ − (cid:1) exp (cid:16) yzz − (cid:17) ( − z θ ) α + ( − z ) α + (cid:16) − | z | (cid:17) α − d λ I ( z ) in view of the explicit expression of the kernel function A α slice in (3.4).Using the generating function for generalized Laguerre polynomials [2,p.288] ( − z ) − α − exp (cid:18) xzz − (cid:19) = ∞ ∑ n = L ( α ) n ( x ) z n , provided | z | <
1, as well as Fejer ’s formula [16, Theorem 8.22.1, p. 198],it is not hard to see that the involved z -function series are uniformly con-vergent on any compact set contained in unit disk. Therefore, direct com-putation yields (cid:102) R αθ ( x , y ) = π Γ ( α ) (cid:90) D (cid:32) ∞ ∑ n = L ( α ) n ( x ) θ n z n (cid:33) (cid:32) ∞ ∑ m = L ( α ) m ( y ) z m (cid:0) − | z | (cid:1) α − (cid:33) d λ ( z )= π Γ ( α ) ∞ ∑ n = ∞ ∑ m = θ n L ( α ) n ( x ) L ( α ) m ( y ) (cid:90) D z n z m (cid:0) − | z | (cid:1) α − d λ ( z )= ∞ ∑ n = n ! Γ ( n + + α ) θ n L ( α ) n ( x ) L ( α ) n ( y ) (3.12)for | θ z | < | θ | ≤ | z | <
1. This provides theexpansion series of the restriction of (cid:102) R αθ ( x , y ) to any B I . For | θ | <
1, werecognize the Hille–Hardy identity (2.5) for Laguerre polynomials. Thus,we have (cid:102) R αθ ( x , y ) = ( − θ ) (cid:18) xy θ (cid:19) α /2 I α (cid:18) θ − θ √ xy (cid:19) exp (cid:18) − θ ( x + y ) − θ (cid:19) (3.13)for | θ | < θ / ∈ R . This leads to (2.7) by considering the kernel func-tion (cid:102) R αθ ( x , y ) x α e − x . The right-hand side in (3.13) is clearly a slice regularfunction in θ ∈ B for x , y being reals. The extension of (3.13) to the wholeunit open ball B relies on the Identity Principle for left slice regular func-tions [8], since both sides of (3.13) are left slice regular and coincide atleast on the upper half unit ball. To conclude, we need only to examinethe validity of the closed expression in the right-hand side of (3.13) for theexpansion of (cid:102) R αθ ( x , y ) with remains valid when | θ | = θ (cid:54) =
1. Thiscan be handled by fixing θ such as and let ε ∈ (
0, 1 ) , so that (3.13) holdstrue for | εθ | <
1, and next sending ε to 1 − , at least formally. This can berigorously justified making use of test functions and classical argumentfrom the Schwartz theory of distributions. (cid:3) Remark 3.6.
By taking θ = − with √ θ = i in (3.11) , we recover the classicalFourier-Bessel transform [14, 11] ( H α ψ ) ( y ) : = (cid:90) ∞ u J α ( yu ) ψ ( u ) dufor ψ ∈ L ( R + ) , where J α is the Bessel function. Indeed, by setting (cid:102) L α = (cid:103) L α − and making the change of variable u = x and the function ψ ( u ) = x α /2 e − x /2 ϕ ( x ) = u α e − u /2 ϕ ( u ) we get (cid:102) L α ϕ ( y ) = e y /2 i α y α (cid:90) ∞ u α + I α ( iyu ) e − u /2 ϕ ( u ) du = e y /2 y α ( H α ψ ) ( y ) . The last equality follows since I α ( x ) = i − α J α ( ix ) . Remark 3.7.
The considered family of QFrFT on the real half-line appears em-bedded in a strongly continuous one-parameter group of unitary operators thequaternionic context. Moreover, it is continuous and interpolates continuouslythe identity operator ( θ = ) to the Hankel transform [2, p. 216] correspondingto θ = − . Remark 3.8.
The considered transform can be used to reintroduce the hyper-holomorphic Bergman space A α slice ( B ) in (3.2) as well as some of their specificgeneralization in the context of slice regular functions on the unit quaternionicball by considering the dual transform of θ (cid:55)−→ (cid:102) L αθ ϕ ( y ) , for fixed y ∈ ( + ∞ ) .For the limit case of y = , the last transform is nothing than the Bargmanntransform in (3.1) . R EFERENCES [1] Almeida L.B., The fractional Fourier transform and time-frequency representation, IEEE trans. sig.proc., 42 (1994) 3084-3091.[2] Andrews G.E., Askey R., Roy R. Special functions. Encyclopedia of Mathematics and its Applications,71. Cambridge University Press: Cambridge; 1999.[3] Bargmann V., On a Hilbert space of analytic functions and an associated integral transform. Comm.Pure Appl. Math. 14 (1961) 187–214.[4] Benahmadi A., Ghanmi A., Non-trivial 1-d and 2-d Segal–Bargmann transforms.
Integral TransformsSpec. Funct.
30 (2019) 547–563.[5] Celeghini1 E., Gadella M., del Olmo M.A., Hermite functions, Lie groups and Fourier analysis. En-tropy 20 (2018), no. 11, Paper No. 816, 14 pp.[6] Colombo F., Sabadini I., Struppa D.C., Entire slice regular functions. Springer, Briefs in Mathematics,Springer International Publishing 2016.[7] Elkachkouri A., Ghanmi A., The hyperholomorphic Bergman space on B R :Integral representation andasymptotic behavior. Complex Anal. Oper. Theory 12 (2018), no. 5, 1351–1367. [8] Gentili G., Stoppato C., The open mapping theorem for regular quaternionic functions . Ann. Sc. Norm.Super. Pisa Cl. Sci. (5) 8 (2009), no. 4, 805–815.[9] Ghanmi A., Zine k., Bicomplex analogs of Segal-Bargmann and fractional Fourier transforms. Adv.Appl. Clifford Algebr. 29 (2019) (4) Paper No. 74, 20 pp.[10] H ¨ormander L., Symplectic classification of quadratic forms, and general Mehler formulas, Math. Z.219 (1995), no. 3, 413–449.[11] Kerr F.H., A fractional power theory for Hankel transforms. J. Math. Anal. Appl. 158 (1991) 114-123[12] Kilbas A.A., Srivastava H.M., Trujilo J.J., Theory and applications of Fractional Differential Equations,Amsterdam, Netherlands, Elsevier, 2006.[13] Namias V., The fractional order Fourier transform and its application to quantum mechanics. J. Inst.Maths. Applics., 25 (1980) 241265[14] Namias V., Fractionalization of Hankel transforms. J. Inst. Math. Appl. 26 (1980), no. 2, 187–197.[15] Ozaktas H.M., Zalevsky A., Kutay M.A., The fractional Fourier transform with Applications in opticsand signal processing, Wiley, New York, 2001.[16] Szeg ¨o G., Orthogonal Polynomials, Colloquium Publications, vol. 23, 4th ed., American MathematicalSociety, Providence, R.I., 1975.[17] Toft J. Images of function and distribution spaces under the Bargmann transform, J. Pseudo-Differ.Oper. Appl. 8 (2017) 83–139[18] Wiener N., Hermitian polynomials and Fourier Analysis, J. Math Phys, 8 (1929) 70–73, Collectedworks Vol. II, pp. 914–918.A
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