Harmonic Analysis on the Möbius Gyrogroup

In this paper we propose to develop harmonic analysis on the Poincaré ball Btn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {B}}_{t}^{n}}$$\end{document}, a model of the n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document}-dimensional real hyperbolic space. The Poincaré ball Btn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {B}}_{t}^{n}}$$\end{document} is the open ball of the Euclidean n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$n$$\end{document}-space Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^n$$\end{document} with radius t>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t >0$$\end{document}, centered at the origin of Rn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^n$$\end{document} and equipped with Möbius addition, thus forming a Möbius gyrogroup where Möbius addition in the ball plays the role of vector addition in Rn.\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^n.$$\end{document} For any t>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t>0$$\end{document} and an arbitrary parameter σ∈R\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\sigma \in \mathbb {R}$$\end{document} we study the (σ,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\sigma ,t)$$\end{document}-translation, the (σ,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\sigma ,t)$$\end{document}-convolution, the eigenfunctions of the (σ,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\sigma ,t)$$\end{document}-Laplace–Beltrami operator, the (σ,t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(\sigma ,t)$$\end{document}-Helgason Fourier transform, its inverse transform and the associated Plancherel’s Theorem, which represent counterparts of standard tools, thus, enabling an effective theory of hyperbolic harmonic analysis. Moreover, when t→+∞\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$t \rightarrow +\infty $$\end{document} the resulting hyperbolic harmonic analysis on Btn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {B}}_{t}^{n}}$$\end{document} tends to the standard Euclidean harmonic analysis on Rn,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbb {R}^n,$$\end{document} thus unifying hyperbolic and Euclidean harmonic analysis. As an application we construct diffusive wavelets on Btn\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {B}}_{t}^{n}}$$\end{document}.


Introduction
Möbius addition, ⊕, in the ball B n t = {x ∈ R n : x < t} plays a role analogous to that of vector addition, +, in the Euclidean n-space R n , giving rise to the Möbius gyrogroup (B n t , ⊕), which is analogous to the Euclidean group (R n , +) [20,23,25]. Möbius gyrogroup turn out to be isomorphic, in the gyrovector sense, to corresponding Einstein gyrogroup [23,Sect. 6.19].
The gyrogroup structure is a natural extension of the group structure, discovered in 1988 by Ungar in the context of Einstein's velocity addition law [18]. The term gyrogroup was coined in 1991 [19,23,Sect. 1.2], following which it has been extensively studied by Ungar and others; see, for instance, [9,10,16,21,23,24,26,27], in the context of abstract algebra, non-Euclidean geometry, mathematical physics, and quantum information and computation.
Möbius addition in the open unit disc D = {z ∈ C : |z| < 1} is the well-known binary operation a ⊕ z = (a + z)(1 +āz) −1 , a, z ∈ D, given by a fractional linear transformation. Möbius addition is neither commutative nor associative, but it is both gyrocommutative and gyroassociative under gyrations defined by gyr [a, b] = (1 + ab)/(1 +āb), a, b ∈ D. The generalisation to higher dimensions of the Möbius addition is done by considering Möbius transformations on the ball.
Möbius transformations in R n were studied by Vahlen in his seminal but almost forgotten paper [29]. Their matricial representation and general properties were rediscovered by Ahlfors [1,2] almost seventy years later, and independently by Hua [13]. Ahlfors noticed that changing the role of the variables in the Möbius transformation on the ball gives the same Möbius transformation up to a specific orthogonal transformation [1]. This orthogonal transformation (denoted by Ungar as Ahlfors rotation) plays a central role in gyrogroup theory and hyperbolic geometry [22,23] and gives rise to the gyration operator. Moreover, it can be regarded as an analogue of the Thomas precession in the theory of special relativity for the Beltrami-Klein model of hyperbolic geometry governed by Einstein's addition of velocities. By incorporating the gyration operator gyr [a, b] in the algebraic structure, gyrogroup theory repairs the breakdown of associativity and commutativity. In parallel to these advances, Clifford algebras appear as an adequate tool for representation of Möbius transformations (see e.g. [4,28]). For instance, using the Clifford algebra representation the gyration operator has an explicit spin representation in the case of the Möbius gyrogroup, which in turn allows the construction of explicit factorisations of the ball with respect to Möbius addition [8,9].
In this paper we propose to study hyperbolic gyroharmonic analysis on the Poincaré ball B n t . With this aim in mind we generalize the results obtained in [17]. The goal of our study is two-fold: first, to understand how the gyration operator affects harmonic analysis on the ball; second, to set the stage for an operator calculus in the framework of wavelet analysis, Gabor analysis, and diffusive wavelets on the ball using the algebraic structure of the Möbius gyrogroup.
In our approach we consider a generalised Laplace-Beltrami operator on the ball depending on the radius t ∈ R + and on an additional parameter σ ∈ R. This operator is a variation of the common Laplace-Beltrami operator, or conformal Laplacian on the unit ball, which plays an important role in scattering and potential theory. It has connections with other equations like the Weinstein equation (see e.g. [17] and references therein). Using the gyrolanguage we prove new theorems like a Young's inequality for the (σ, t)-convolution (Theorem 1), the gyrotranslation invariance of the (σ, t)-convolution (Theorem 2), the gyroassociative law of the (σ, t)-convolution (Theorem 3), and the generalised convolution theorem with respect to the (σ, t)-Helgason Fourier transform (Theorem 5). Each of those theorems involves the gyration operator in a natural way. In the context of the unit ball [17] and symmetric spaces [11,12] Theorems 3 and 5 are known only in the radial case. In contrast, the gyrogroup theoretic techniques used in this paper enable us to remove the radial condition. It is interesting to explore the translation of these theorems to other models of hyperbolic geometry as, for instance, the upper half space or the hyperboloid, and more generally, Riemannian globally symmetric spaces of noncompact type, but we will not address these problems here.
The paper is organised as follows. In Sect. 2 we present the Möbius addition in the ball B n t and its properties. Sections 3 and 4 are dedicated to the study of the (σ, t)translation and the (σ, t)-convolution. In Sect. 5 we construct the eigenfunctions of the generalised Laplace-Beltrami operator and study the associated (σ, t)-spherical functions. In Sect. 6 we define the (σ, t)-Helgason Fourier transform, which is the relativistic counterpart of the Euclidean Fourier transform. In Sect. 7 we obtain the inversion formula for the (σ, t)-Helgason Fourier transform, the Plancherel's Theorem, and show that in the limit t → +∞ we recover the inverse Fourier transform and Plancherel's Theorem in Euclidean harmonic analysis. Finally, in Sect. 8 we construct diffusive wavelets on B n t arising from the heat kernel associated to the generalised Laplace-Beltrami operator σ,t . Two appendices, A and B, concerning all necessary facts on spherical harmonics and Jacobi functions, are found at the end of the article.

Möbius Addition in the Ball
The Poincaré ball model of n-dimensional hyperbolic geometry is the open ball B n t = {x ∈ R n : x < t} of R n , endowed with the Poincaré metric The Poincaré metric is normalised so that in the limit case t → +∞ one recovers the Euclidean metric. The group M(B n t ) of all conformal orientation preserving transformations of B n t is given by the mappings K ϕ a , where K ∈ SO(n) and ϕ a are Möbius transformations on B n t given by (see [2,9]) where a, x ∈ B n t , a, x being the usual scalar product in R n , and x being the Euclidean norm. Furthermore, ax stands for the Clifford multiplication which we now recall. The Clifford algebra C 0,n over R n is the associative real algebra generated by R n and R subject to the relation x 2 = − x 2 , for all x ∈ R n . Therefore, given an orthonormal basis {e j } n j=1 of R n we have the multiplication rules: e j e k + e k e j = 0, j = k, and e 2 j = −1, j = 1, . . . , n. Any non-zero vector x ∈ R n is invertible and its inverse is given by The geometric product between two vectors is given by involving the symmetric part 1 2 (x y + yx) = − x, y and the anti-symmetric part 1 2 (x y − yx) := x ∧ y, also known as the outer product. The norm in R n can be extended to C 0,n and then, for two vectors we have x y = x y . This equality is not true for general elements in the Clifford algebra. For more details about the Clifford product and the Clifford norm see [5,9]. In order to endow the manifold B n t with an algebraic structure one defines the Möbius addition as In [9] we proved that (B n t , ⊕) is a gyrogroup, i.e., the following properties hold: (P1) There is a left identity: 0 ⊕ a = a, for all a ∈ B n t ; (P2) There is a left inverse: ( a) ⊕ a = 0, for all a ∈ B n t ; (P3) Möbius addition is gyroassociative, that is, for any a, b, c ∈ B n Here is the gyration operator [9], which corresponds to a spin rotation induced by an element of the group Spin(n) (double covering group of SO(n)); (P4) The gyroautomorphism gyr We remark that a = −a and Möbius addition (3) corresponds to a left gyrotranslation as defined in [23]. In the limit t → +∞, the ball B n t expands to the whole of the space R n , Möbius addition reduces to vector addition in R n and, therefore, the gyrogroup (B n t , ⊕) reduces to the translation group (R n , +). The Möbius gyrogroup is gyrocommutative since Möbius addition satisfies the property Some useful gyrogroup identities ( [23], pp. 48 and 68) that will be used in this article are Properties (8) and (9) are valid for general gyrogroups while properties (6) and (12) are valid only for gyrocommutative gyrogroups. Combining formulas (9) and (12) with (8) we obtain new identities Möbius transformations (1) satisfy the following useful relations and In the special case when n = 1, the Möbius gyrogroup becomes a group since gyrations are trivial (a trivial map being the identity map). For n ≥ 2 the gyrosemidirect product [23] of (B n t , ⊕) and Spin(n) gives a group B n t gyr Spin(n) for the operation We remark that this group is a realisation of the proper Lorentz group Spin + (1, n) (double covering group of SO 0 (1, n)). In the limit t → +∞ the group B n t gyr Spin(n) reduces to the Euclidean group E(n) = R n Spin(n). The harmonic analysis presented in this paper is associated to the family of Laplace-Beltrami operators σ,t defined by These operators are considered in [17] for the case of the unit ball. The case σ = 2 − n and t = 1 corresponds to the conformally invariant operator associated to the Poincaré disk model. In the limit t → +∞ the operator σ,t reduces to the Laplace operator in R n . Therefore, harmonic analysis associated to σ,t in B n t provides a link between hyperbolic harmonic analysis and the classic harmonic analysis in R n .

Definition 1
For a function f defined on B n t and a ∈ B n t we define the (σ, t)- with The multiplicative factor j a (x) is a positive function and agrees with the Jacobian of the transformation ϕ −a (x) = (−a) ⊕ x when σ = n + 2. In the case σ = 2 − n the (σ, t)-translation reduces to τ a f (x) = f ((−a) ⊕ x). Moreover, for any σ ∈ R, we obtain in the limit t → +∞ the Euclidean translation operator τ a f ( Lemma 1 For any a, b, x, y ∈ B n t the following relations hold Proof In the proof we use the following properties of the Clifford product: Identities (19)- (21) can be easily verified by definition. Now we prove equality (22): Equality (23) follows from (16): (20) and (23) since we have

Corollary 1 Let f be a radial function defined on
Before we prove that the generalised Laplace-Beltrami operator σ,t commutes with (σ, t)-translations we present a representation formula for the operator σ,t using the Laplace operator in R n .
Proof Let a ∈ B n t and denote by T 1 , . . . , T n the coordinates of the mapping ϕ a (x). Then by the chain rule we have we obtain by putting Therefore, we get .
Proof Using (38) we have (4)) and (10)) then together with the invariance of under the group SO(n), (23) and (19) we obtain For studying some L 2 -properties of the invariant Laplace σ,t and the (σ, t)translation we consider the weighted Hilbert space where dx stands for the Lebesgue measure in R n . For the special case σ = 2 − n we recover the invariant measure associated to the Möbius transformations ϕ a (x).
Proof By definition we have Making the change of variables (−a)⊕ x = z, which is equivalent by (7) to x = a ⊕ z, the measure becomes (16), (18)).
Corollary 2 For f, g ∈ L 2 (B n t , dμ σ,t ) and a ∈ B n t we have From Corollary 2 we see that the (σ, t)-translation τ a is an unitary operator in L 2 (B n t , dμ σ,t ) and the measure dμ σ,t is translation invariant only for the case σ = 2 − n.
An important property of the Laplace operator in R n is that it is a self-adjoint operator. The same holds for the hyperbolic operator σ,t due to the representation formula (38) (see [17] for the proof in the case t = 1).

The (σ, t)-Convolution
In this section we define the (σ, t)-convolution of two functions, we study its properties and we establish the respective Young's inequality and gyroassociative law. In the limit t → +∞ both definitions and properties tend to their Euclidean counterparts.

Definition 2
The (σ, t)-convolution of two measurable functions f and g is given by The (σ, t)-convolution is commutative, i.e., f * g = g * f. This can be seen by (39) and a change of variables z → −y. It is well defined only for σ < 1 as the next proposition shows.
where g(x) := ess sup y∈B n t g(gyr [y, x]x), for any x ∈ B n t .
General case: Let 1 ≤ q ≤ ∞ and g ∈ L q (B n t , dμ σ,t ). Considering the linear operator T defined by T g ( f ) = f * g we have by the previous cases ||g|| q || f || p , i.e. T : L p → L ∞ with 1/ p + 1/q = 1. By the Riesz-Thorin interpolation theorem we obtain dμ σ,t ) and g ∈ L q (B n t , dμ σ,t ) a radial function. Then, Remark 1 For σ = 2 − n and taking the limit t → +∞ in (45) we recover Young's inequality for the Euclidean convolution in R n since in the limit g = g.
Another important property of the Euclidean convolution is its translation invariance. In the hyperbolic case the convolution is gyrotranslation invariant.
In Theorem 2 if g is a radial function then we obtain the translation invariant property τ a ( f * g) = (τ a f ) * g. The next theorem shows that the (σ, t)-convolution is gyroassociative.

gyr [y, x]x))(y) * a h(y))(a).
Corollary 4 If f, g, h ∈ L 1 (B n t , dμ σ,t ) and g is a radial function then the (σ, t)convolution is associative. i.e., From Theorem 3 we see that the (σ, t)-convolution is associative up to a gyration of the argument of the function g. However, if g is a radial function then the corresponding gyration is trivial (that is, it is the identity map) and therefore the (σ, t)-convolution becomes associative. Moreover, in the limit t → +∞ gyrations reduce to the identity, so that formula (50) becomes associative in the Euclidean case. If we denote by L 1 R (B n t , dμ σ,t ) the subspace of L 1 (B n t , dμ σ,t ) consisting of radial functions then, for σ < 1, L 1 R (B n t , dμ σ,t ) is a commutative associative Banach algebra under the (σ, t)convolution.

Eigenfunctions of σ,t
We begin by defining the (σ, t)-hyperbolic plane waves which are the relativistic counterpart of the Euclidean plane waves and proceed with the study of its properties.

Definition 3
For λ ∈ C, ξ ∈ S n−1 , and x ∈ B n t we define the functions e λ,ξ ;t by (51)

Proposition 5
The function e λ,ξ ;t is an eigenfunction of σ,t with eigenvalue Proof Since and ∇e λ,ξ ;t (x) = −(1−σ +iλt) 1− then we have, by straightforward computations, The (σ, t)-hyperbolic plane waves e λ,ξ ;t (x) converge in the limit t → +∞ to the Euclidean plane waves e i x,η , where η = λξ ∈ R n , for λ ∈ R. For x ∈ R n , choose t 0 > 0 such that x ∈ B n t 0 . Then x ∈ B n t for all t > t 0 and, moreover, Letting t → +∞ we observe that y tends to x and lim t→+∞ e λ,ξ ;t (x) = lim with η = λξ ∈ R n . Moreover, the eigenvalues of σ,t converge to −λ 2 = − η 2 , which are the eigenvalues of the Laplace operator in R n associated to the eigenfunctions e i x,η . In the Euclidean case given two eigenfunctions e i x,λξ and e i x,γ ω , λ, γ ∈ R, ξ, ω ∈ S n−1 of the Laplace operator with eigenvalues −λ 2 and −γ 2 respectively, the product of the two eigenfunctions is again an eigenfunction of the Laplace operator with eigenvalue −(λ 2 + γ 2 + 2λγ ξ, ω ). Indeed, Unfortunately, in the hyperbolic case this is no longer true in general. The only exception is the case n = 1 and σ = 1 as the next proposition shows.

Proposition 6
For n ≥ 2 the product of two eigenfunctions of σ,t is not an eigenfunction of σ,t and for n = 1 the product of two eigenfunctions of σ,t is an eigenfunction of σ,t only in the case σ = 1.
Thus, only in the case n = 1 and σ = 1 the product of two eigenfunctions of σ,t is an eigenfunction of σ,t .
We remark that in the case when n = 1 and σ = 1 the hyperbolic plane waves (51) are independent of ξ since they reduce to and, therefore, the exponential law is valid, i.e., e λ;t (x)e γ ;t (x) = e λ+γ ;t (x).
In the Euclidean case the translation of the Euclidean plane waves e i x,λξ decomposes into the product of two plane waves one being a modulation. In the hyperbolic case we have an analogous result for the (σ, t)-translation of the (σ, t)-hyperbolic plane waves but it appears a Möbius transformation acting on S n−1 as the next proposition shows. we see, by straightforward computations, that F satisfies the following hypergeometric equation: The Finally, we prove the addition formula for the (σ, t)-spherical functions.
The second equality follows from the fact that φ λ;t is an even function of λ, i.e., φ λ;t = φ −λ;t .

The (σ, t)-Helgason Fourier Transform
Definition 5 For f ∈ C ∞ 0 (B n t ), λ ∈ C and ξ ∈ S n−1 we define the (σ, t)-Helgason Fourier transform of f as Remark 3 If f is a radial function i.e., f (x) = f ( x ), then f (λ, ξ ; t) is independent of ξ and reduces by (57) to the so called (σ, t)-spherical Fourier transform defined by Moreover, by (52) we recover in the Euclidean limit the Fourier transform in R n .
Since σ,t is a self-adjoint operator and by Proposition 5 we obtain the following result.
Now we study the hyperbolic convolution theorem with respect to the (σ, t)-Helgason Fourier transform. We begin with the following lemma.
Corollary 5 Let f, g ∈ C ∞ 0 (B n t ) and g radial. Then Since in the limit t → +∞ gyrations reduce to the identity and (−y) ⊕ ξ reduces to ξ , formula (67) converges in the Euclidean limit to the well-know convolution Theorem: Next proposition shows that the Fourier coefficients of a given function f ∈ C ∞ 0 (B n t ) can be related with the (σ, t)-convolution.

Inversion of the (σ, t)-Helgason Fourier Transform and Plancherel's Theorem
We obtain first an inversion formula for the radial case, that is, for the (σ, t)-spherical Fourier transform and then we will derive a general inversion formula for the (σ, t)-Helgason Fourier transform. In the sequel C ∞ 0,R (B n t ) denotes the space of all radial C ∞ functions on B n t with compact support and C n,t,σ = 1 2 −1+2σ t n−1 π A n−1 .
By (64), Fubini's Theorem, and the change of variables K y → z we have Since f (x) = f x (0) it follows from (74) any n ∈ N and λ ∈ R + (see [3]), we see that in the Euclidean limit the (σ, t)-Helgason This theorem provides a generalisation of Theorem 2.3 in [15] for arbitrary t ∈ R + . From [15] and considering t ∈ R + arbitrary we have the following asymptotic behavior of φ α,β λt for Im(λ) < 0 :