**1. Introduction**

It is well known that the classical Fourier transform refers to the decomposition of a function belonging to an appropriate space into exponentials, which can be viewed as the irreducible unitary representations of the additive group of the real numbers. However, in the current most popular interpretation, particularly in connection with non-commutative groups, the phrase *harmonic analysis* has lost its original function-theoretic meaning and now it generally refers not to functions but to representations. It thus becomes natural to regard irreducible representations as the basic building blocks of the theory in place of exponential functions [1]. There are however examples in the theory of the semi-simple non-compact Lie groups where the classical setup prevails, in the sense that one can find a class of functions which play a role similar to that played by the exponentials on the real line. A typical example thereof is the group SU(1, <sup>1</sup>): hereafter, we shall work at the level of homogeneous spaces associated with this group and, accordingly, we shall study the spherical functions that can be constructed in these spaces.

Within this framework, a fundamental role is played by the Helgason construction of the so-called *non-Euclidean Fourier analysis* [2]. The working ambient is the symmetric space SU(1, 1)/ SO(2), i.e., the non-Euclidean disk. By the use of the Poisson kernel, the analog of the plane waves are constructed in the case of the hyperbolic disk and, successively, an integral representation of the conical functions *P*− 1 2 <sup>+</sup>i*μ*(cosh *r*) (i.e., the Legendre functions of the first kind with index (−<sup>1</sup> 2 + <sup>i</sup>*μ*), *μ* ∈ R) is derived in terms of these *hyperbolic waves*. It can thus be proved for these functions a product formula, which corresponds to the classical product formula of the exponentials. Finally, a Fourier transform on the non-Euclidean disk can be set up, which is exactly the tool analogous to the classical Fourier transform. In this connection, it is worth recalling that the conical functions can be associated with the

principal series of the irreducible unitary representation of the group SU(1, <sup>1</sup>), which acts transitively on the hyperbolic disk.

Consider now an isotropic cone in R3. We focus on the two-sheeted hyperboloid interior to the cone. By stereographic projection, the upper sheet of the two-sheeted hyperboloid can be mapped into the interior of the unit disk. Since the connected component of the two-sheeted hyperboloid is the homogeneous space SO0(1, 2)/ SO(2), the non-Euclidean Fourier analysis coincides with the harmonic analysis on SO0(1, 2)/ SO(2). We can thus define and study the Fourier transform on the two-sheeted hyperboloid.

Harmonic analysis can be studied also on the one-sheeted hyperboloid, which is a pseudo-Riemannian symmetric space SO0(1, 2)/ SO0(1, <sup>1</sup>). It is then possible to construct on this symmetric space spherical functions, which turn out to be the Legendre functions of the second kind. A peculiar feature of this space is that it can be equipped with the partial ordering associated with the light cone in R3: namely, *x y* ⇔ (*x* − *y*) belongs to the closed future cone of R3. The one-sheeted hyperboloid equipped with this ordering relation is a *causal symmetric space* [3]. Accordingly, we can introduce a *Volterra algebra* of kernels, i.e., kernels *<sup>K</sup>*(*<sup>x</sup>*, *y*) whose support is contained in the set Γ . = {(*<sup>x</sup>*, *y*) ∈ *X*2 × *X*2 : *x y*}, where *X*2 is the one-sheteed hyperboloid. A kernel *K* is said to be invariant under *G* ≡ SO0(1, 2) if for any *g* ∈ *G*: *<sup>K</sup>*(*gx*, *gy*) = *<sup>K</sup>*(*<sup>x</sup>*, *y*), (*<sup>x</sup>*, *y*) ∈ Γ. An invariant Volterra kernel *K* can be identified with a function *f* on *G* through the equality: *<sup>K</sup>*(*ge*2,*e*2) = *f*(*g*), where *g* ∈ *G* and *e*2 = (0, 0, 1) (see next Figure 1) is the point which features *he*2 = *e*2, with *h* ∈ SO0(1, 1) [4–7]. Successively, the *spherical Laplace transform* for this class of functions can be defined. This transform, as the ordinary Laplace transform, is holomorphic in a half-plane and, in the specific case of the spherical Laplace transform, this analyticity property follows from the analyticity property of the Legendre functions of the second kind *Qλ*(·), which are holomorphic in the half-plane C(+) (−<sup>1</sup>) . = {*λ* ∈ C : Re *λ* > <sup>−</sup><sup>1</sup>}. These latter functions are indeed the spherical functions on ordered symmetric spaces in the sense of Faraut et al. [7].

**Figure 1.** Horocyclic fibration of the one-sheteed hyperboloid *X*<sup>+</sup> 2 .

Another question closely related to this type of problems is the holomorphic extension associated with Fourier–Legendre expansions of the type: 14*π* ∑∞*<sup>n</sup>*=<sup>0</sup>(<sup>2</sup>*<sup>n</sup>* + <sup>1</sup>)*anPn*(cos *<sup>θ</sup>*), where *Pn*(·) denotes the Legendre polynomials. In this paper, we prove that, if the sequence {(*n* + <sup>1</sup>)<sup>2</sup>*an*}<sup>∞</sup>*n*=<sup>0</sup> satisfies a suitable Hausdorff-type condition [8], then the function *f*(cos *<sup>θ</sup>*), to which the Fourier–Legendre series converges in the interval (−1, <sup>1</sup>), admits a holomorphic extension to the complex cos *θ*-plane cut along the semi-axis [1, <sup>+</sup>∞). The first result we prove is a basic feature of the holomorphic extension associated with Fourier–Legendre series: the *dual analyticity property*. To classes of functions which are holomorphic in the cos *θ*-plane cut along [1, <sup>+</sup>∞), there correspond classes of functions, denoted *a*˜(*λ*) (*λ* ∈ C), which are expressed as a spherical Laplace transform, holomorphic in the half-plane Re *λ* > −12 , of Carlsonian-type (i.e., of suitable exponential growth; see next Theorem 1) and, therefore, unique interpolants of the coefficients *an* of the Fourier-Legendre series, i.e., *<sup>a</sup>*˜(*λ*)|*<sup>λ</sup>*=*<sup>n</sup>* = *an* (*n* = 0, 1, 2, . . .).

At this point, two strictly related problems emerge. On the one hand, the possibility of connecting, via analytic continuation, the completeness of the spherical functions for SO(3) to the completeness of the corresponding expansion of SL(2, <sup>R</sup>), which is the group acting transitively on the upper half-plane, where the model of the non-Euclidean geometry can be realized, while, on the other hand, continuing the unitary representations of SO(3) to give the unitary representations of SL(2, <sup>R</sup>). The first problem has been initially tackled by Stein and Wainger [9], while, for what concerns the second one, the reader is referred to the work of R. Hermann [10]. However, both of these problems require linking the spherical Laplace transform to the non-Euclidean Fourier analysis. When we consider the ordinary Laplace transform of a function *f* ∈ *L*<sup>1</sup>(0, <sup>+</sup>∞) ∩ *L*<sup>2</sup>(0, <sup>+</sup>∞),

$$\widetilde{f}(\lambda) = \int\_0^{+\infty} e^{-\lambda v} f(v) \, \mathrm{d}v \qquad (\lambda \in \mathbb{C}, \mathrm{Re}\,\lambda > 0),$$

we obtain for *λ* = i*μ* (*μ* ∈ R):

$$\widetilde{f}(\mathbf{i}\mu) = \int\_{-\infty}^{+\infty} e^{-i\mu v} f(v) \,\mathrm{d}v \qquad (\mu \in \mathbb{R})\_{\prime}$$

which is precisely the Fourier transform of *f*(*v*), it being well-defined in view of the assumption *f* ∈ *L*<sup>1</sup>(0, <sup>+</sup>∞). Since *f* ∈ *L*<sup>1</sup>(0, <sup>+</sup>∞) ∩ *L*<sup>2</sup>(0, <sup>+</sup>∞), the Fourier transform can be inverted to recover *f*(*v*) in terms of *f* , (<sup>i</sup>*μ*) through the *inverse Fourier transform*, which converges to *f*(*v*) as a limit in the mean order two. Correspondingly, we pass from the non-unitary representation *e*<sup>−</sup>*λ<sup>v</sup>* (*λ* ∈ C, Re *λ* > 0, *μ* ≡ Im *λ* ∈ R) to the unitary irreducible representation *e*<sup>−</sup>i*μ<sup>v</sup>* (*μ*, *v* ∈ R) that we mentioned at the beginning of the Introduction. This connection can be extended. One of our results consists indeed of proving that the spherical Laplace transform reduces to the non-Euclidean Fourier transform at Re *λ* = −12 , which is precisely the value corresponding to the principal series of the unitary representations of the group SU(1, <sup>1</sup>), which acts transitively on the hyperbolic disk. This result has been obtained by establishing a bridge between the harmonic analysis on the one-sheeted hyperboloid and the harmonic analysis on the two-sheeted hyperboloid.

The harmonic analysis in causal symmetric spaces has been a subject of growing interest in the last three decades, and the research on these topics has flowed in various directions. Some papers have been devoted to the proof of the Paley–Wiener theorem for spherical Laplace transform [11,12] and others have treated the inversion problem extended up to the so-called Θ-transform [13]. In [14], Bertram has studied the compact symmetric space and its non-compact dual, both realized as real forms of their common complexification; this analysis has been performed by exploring the Ramanujan garden and by the use of the so-called *Master Theorem*. Working in a different direction, Gindinkin and Krötz [15] have studied the complex crown of Riemannian symmetric spaces and non-compactly causal symmetric spaces.

This allowed them to prove the conjecture that every non-compactly causal symmetric space occurs as a component of a distinguished boundary of some complex crown [15]. Finally, it is worth recalling that the question of relating the harmonic analysis of different real forms of a complex symmetric space has been studied also in the context of scattering theory and resonances [16–20].

The paper is organized as follows. In Section 2, we study the holomorphic extension associated with the Fourier–Legendre expansion by means of the spherical Laplace transform. We are thus led to develop the harmonic analysis on the complex one-sheeted hyperboloid *X*(*c*) 2 , which contains as submanifolds either the Euclidean sphere (iR × R<sup>2</sup>) ∩ *X*(*c*) 2 on which the Fourier–Legendre expansion can be developed, and the real one-sheeted hyperboloid, which contains the support of the cut. In Section 3, we analyze the relationship between the spherical Laplace transform and the *non-Euclidean Fourier transform*. For this purpose, we consider a real two-sheeted hyperboloid, and, in this geometrical setting, we can recover the non-Euclidean Fourier transform in the sense of Helgason. Finally, in Section 4, some conclusions will be drawn.

### **2. Holomorphic Extension Associated with the Fourier–Legendre Expansion and the Spherical Laplace Transform**

### *2.1. The Complex One-Sheeted Hyperboloid*

In the space C<sup>3</sup> of the variable *z* = (*<sup>z</sup>*0, *z*1, *<sup>z</sup>*2), we consider the complex quadric *X*(*c*) 2with equation

$$-z\_0^2 + z\_1^2 + z\_2^2 = 1\_\prime$$

which is a one-sheeted complex hyperboloid. Next, we introduce two systems of coordinates: polar and horocyclic coordinates.
