3.4.1. Preparatory Lemmas

**Lemma 1.** *(i) The Poisson kernel*

$$\mathbb{P}(z,b) = \frac{1-|z|^2}{1+|z|^2 - 2|z|\cos(\psi-\phi)}\qquad \left(z=|z|e^{i\psi'}, \psi'=\frac{\pi}{2}-\psi; b=e^{i\phi'}, \phi'=\frac{\pi}{2}-\phi\right),$$

*is constant on each horocycle Hb*(*z*) *with normal b and passing through z* ∈ *D.*

*(ii) The function*

$$\left[\mathbb{P}(z,b)\right]^{\lambda} = \left[\frac{1-|z|^2}{1+|z|^2-2|z|\cos(\psi-\phi)}\right]^{\lambda} \qquad (\lambda \in \mathbb{C}),\tag{64}$$

*is an eigenfunction of the Laplace–Beltrami operator on the hyperbolic disk D, corresponding to the eigenvalue λ*(*λ* − 1)*.*

*(iii) The hyperbolic waves (horocyclic waves) are represented by the following expression:*

$$\epsilon^{\lambda \langle z, b \rangle} = \left[ \frac{1 - |z|^2}{1 + |z|^2 - 2|z| \cos(\psi - \phi)} \right]^{\lambda} \qquad (\lambda \in \mathbb{C}), \tag{65}$$

*where z*, *b is the signed non-Euclidean distance between the center of the unit disk D and the horocycle Hb*(*z*) *with normal b that passes through z* ∈ *D.*

*(iv) The following equality holds:*

$$\left[\frac{1-|z|^2}{1+|z|^2-2|z|\cos(\psi-\phi)}\right]^\lambda = \frac{1}{[\cosh v - \sinh v \cos(\psi-\phi)]^\lambda} \qquad (\lambda \in \mathbb{C}).\tag{66}$$

**Proof.** (i) The level lines of the Poisson kernel <sup>P</sup>(*<sup>z</sup>*, *b*) are the circles tangent from within to the unit circle at the point *b* = *e*i*φ* ; when interpreted in a non-Euclidean fashion, they represent horocycles *Hb*(*z*) with normal *b* (p. 7, [26]).

(ii) The Laplace–Beltrami operator Δ*D* on the non-Euclidean unit disk *D* is given by [2,27]

$$
\Delta\_D = \frac{1}{4} \left[ 1 - \left( \mu\_1^2 + \mu\_2^2 \right) \right]^2 \left( \frac{\partial^2}{\partial \mu\_1^2} + \frac{\partial^2}{\partial \mu\_2^2} \right) \dots
$$

For *λ* ∈ C, a direct computation gives [2,27]

$$
\Delta\_D \left[ \mathbb{P}(z, b) \right]^\lambda = \lambda (\lambda - 1) \left[ \mathbb{P}(z, b) \right]^\lambda \qquad (\lambda \in \mathbb{C}).\tag{67}
$$

(iii) In the Euclidean case, the function *x* → *<sup>e</sup>*i*k*(*<sup>x</sup>*,*<sup>ω</sup>*), *k* ∈ R, *ω* ∈ <sup>S</sup>(*<sup>n</sup>*−<sup>1</sup>), *x* ∈ R*<sup>n</sup>*, represents a plane wave with normal *ω*: it is an eigenfunction of the Laplacian in R*n* and is constant on every hyperplane perpendicular to *ω*. In the case of the non-Euclidean disk *D*, the geometric analog of the plane wave is the function represented by Equation (65) [2,27]. In fact, it is an eigenfunction of the Laplace–Beltrami operator on *D*, as proved by statement (ii) (see (67)). Let *z*min = |*<sup>z</sup>*min|*e*i*<sup>χ</sup>* denote the point on the horocycle *Hb*(*z*) (i.e., the one with normal *b* = *e*i*φ* and passing through *z*) that is closest to the center of *D*. Then, *χ* = *φ* or *χ* = *φ* + *π* depending on whether the origin of *D* lies, respectively, outside or inside the horocycle *Hb*(*z*). Therefore, from the definition of *z*, *b*, we have:

$$|\langle z, b \rangle| \doteq \ln \frac{1 + |z\_{\min}|}{1 - |z\_{\min}|} = |\ln \, \mathbb{P}(z\_{\min}, b)| = |\ln \, \mathbb{P}(z, b)|\, \rho$$

the last equality following from statement (i). If we define *z*, *b* .= ln <sup>P</sup>(*<sup>z</sup>*, *b*), then *z*, *b* is indeed constant on each horocycle *Hb*(*z*) and represents the hyperbolic analog of (*<sup>x</sup>*, *<sup>ω</sup>*). Moreover, *z*, *b* is positive if *z* is such that the origin of *D* lies outside the horocycle *Hb*(*z*), whereas it is negative if the origin falls inside.

(iv) Equality (66) follows by plugging |*z*| = tanh(*v*/2) in formula (64).

We now introduce the Spherical Functions <sup>Φ</sup>*λ*(*g*) on *G*/*K* (*g* ∈ *G* = SU(1, <sup>1</sup>), *K* = SO(2)).

**Definition 1.** *The* Spherical Functions *on G*/*K are defined by [28]*

$$\Phi\_{\lambda}(\mathcal{g}) \doteq \int\_{B} \left| \frac{\mathbf{d}(\mathcal{g}^{-1} \cdot b)}{\mathbf{d}b} \right|^{\lambda} \mathbf{d}b \qquad (\mathcal{g} \in \mathcal{G}, \lambda \in \mathbb{C}),$$

*where B* = {*z* : |*z*| = 1} *is the boundary of the non-Euclidean disk D.*

**Lemma 2** (Eymard [28])**.** *The functions* <sup>Φ</sup>*λ*(*g*) *satisfy the following properties:*

*(i)* <sup>Φ</sup>*λ*(*g*) = *<sup>P</sup>*−*<sup>λ</sup>*(cosh *<sup>v</sup>*)*,* (*g* ∈ *G*, *λ* ∈ C, *v* ∈ <sup>R</sup>)*, where <sup>P</sup>*−*<sup>λ</sup>*(·) *are the Legendre functions of the first kind. (ii)* <sup>Δ</sup>*D*Φ*λ*(*g*) = *λ*(*λ* − <sup>1</sup>)<sup>Φ</sup>*λ*(*g*)*,* (*g* ∈ *G*, *λ* ∈ C)*, where* Δ*D is the non-Euclidean Laplace–Beltrami operator. (iii) For λ* = −12 + i*μ (μ* ∈ R*):*

$$\begin{split} \langle \langle ii.a \rangle \quad & P\_{-\frac{1}{2} + i\mu} \langle \cosh v \rangle = P\_{-\frac{1}{2} - i\mu} \langle \cosh v \rangle, \\ \langle iii.b \rangle \quad & P\_{-\frac{1}{2} + i\mu} \langle \cosh v \rangle = \frac{1}{2\pi} \int\_{0}^{2\pi} \left( \frac{1}{\cosh v - \sinh v \cos(\psi - \phi)} \right)^{\frac{1}{2} - i\mu} \, \mathrm{d}\phi \\ &= \frac{1}{2\pi} \int\_{0}^{2\pi} \left( \frac{1 - |z|^{2}}{1 + |z|^{2} - 2|z| \cos(\psi - \phi)} \right)^{\frac{1}{2} - i\mu} \mathrm{d}\phi \\ &= \frac{1}{2\pi} \int\_{\beta} e^{(\frac{1}{2} - i\mu)\langle z, b \rangle} \, \mathrm{d}b \qquad (z \in D). \end{split} \tag{68}$$

### **Proof.** See Ref. [28].

**Remark 6.** *Representation* (68) *of the conical function is attained by varying within the integral the angle φ (or, equivalently, the normal b) in such a way as to span the entire horizon (φ* ∈ [0, <sup>2</sup>*π*]*). This amounts to considering in the non-Euclidean disk all the families of horocycles which are obtained by varying the normal b on the horizon. On the other hand, we have seen in Section 3.2 that only one family of horocycles (the one with tangency point at z* = −i*) results from the fibration of the upper sheet* <sup>2</sup>*X*+2 *of the real two-sheeted hyperboloid. However, from formula* (68)*, we see that the same representation of the conical function can be obtained by keeping φ fixed and varying ψ in the range* [0, <sup>2</sup>*π*]*. Let us recall once again that here we consider only functions of the form F* = *F*(cosh *v*) *and therefore not depending on b (see Remark 4(ii)). Consequently, in the non-Euclidean transform (that will be analyzed in the next subsection), the b-dependence of the integrand derives solely from the integral representation* (68) *of the conical function.*

### 3.4.2. Non-Euclidean Fourier Transform

It is well known that the classical Fourier transform refers to the decomposition of a function, belonging to an appropriate space, into exponentials of the form *e*i*kx* (*k* real), which can also be viewed as the irreducible unitary representations of the additive group of the real numbers. However, the non-Euclidean disk is not a group. Therefore, a straightforward generalization of this viewpoint is not applicable here. Nevertheless, in view of the fact that the functions *<sup>P</sup>*−*<sup>λ</sup>*(cosh *v*) correspond for *λ* = 12 − i*μ* to the principal series of the irreducible unitary representations of the group SU(1, <sup>1</sup>), the exponentials *e*( 12 <sup>−</sup>i*μ*) *<sup>z</sup>*,*b* (*μ* ∈ R) (see statement (iii.b) of Lemma 2) represent the analog of the Euclidean exponentials and play the same role in the non-Euclidean Fourier analysis. We can now state the following classical theorem due to Helgason.

,

**Theorem 4** (Helgason [2,27])**.** *For f* ∈ *C*<sup>∞</sup>*c* (*D*)*, let f denote the Fourier transform* 

$$\widetilde{f}(\mu, b) = \int\_{D} e^{(\frac{1}{2} - \mathfrak{i}\mu)\langle z, b\rangle} f(z) \, \mathrm{d}z \qquad (\mu \in \mathbb{R}, b \in \mathcal{B}), \tag{69}$$

*where* d*z is the invariant surface element on D. Then:*

> *(i) The inverse Fourier transform is given by*

> > ,

$$f(z) = \frac{1}{(2\pi)^2} \int\_{\mathbb{R}^+} \int\_B e^{(\frac{1}{2} + i\mu)\langle z, b\rangle} \widetilde{f}(\mu, b) \,\mu \tanh\pi\mu \,d\mu \,d\nu,\tag{70}$$

*where* d*b is the angular measure on B.*

*(ii) The mapping f* → *f extends to an isometry of the space <sup>L</sup>*<sup>2</sup>(*<sup>D</sup>*, d*z*) *onto the space L*<sup>2</sup>(R<sup>+</sup> × *B*, 12*πμ* tanh *πμ* d*μ* d*b*)*.*

**Proof.** See Refs. [2,27].

**Remark 7.** *(i) Since we are considering functions of the form F* = *F*(cosh *<sup>v</sup>*)*, we can restrict the integrals* (69) *and* (70) *to the case of radial functions [27] that is, to functions f*(*z*) = *F*(*d*(0, *<sup>z</sup>*))*, F even. In fact, since d*(0, *z*) = ln <sup>1</sup>+|*z*| <sup>1</sup>−|*z*| *and* |*z*| = tanh(*v*/2)*, then d*(0, *z*) = *v and f*(*z*) = *<sup>F</sup>*(*v*) = *F*(cosh *<sup>v</sup>*)*. Hence, f*,(*μ*, *b*) *is an even function of μ alone. In view of statements (iii.a) and (iii.b) of Lemma 2 (see also Remark 6), noting that the expression of the invariant surface element* d*z in the coordinates* (*<sup>v</sup>*, *ψ*) *is* d*z* = sinh *v* d*v* d*ψ, and writing f* , (*μ*, *b*) = 2*πF* , (*μ*)*, then formulas* (69) *and* (70) *read*

$$\hat{F}(\mu) = \int\_0^{+\infty} \underline{F}(\cosh \upsilon) \, P\_{-\frac{1}{2} + i\mu}(\cosh \upsilon) \sinh \upsilon \, \mathrm{d}\upsilon,\tag{71a}$$

$$F(v) = \underline{F}(\cosh v) = \int\_0^{+\infty} \hat{F}(\mu) \, P\_{-\frac{1}{2} + i\mu}(\cosh v) \, \mu \tanh \pi \mu \, \mathrm{d}\mu,\tag{71b}$$

*which coincide with formulas* (54)*, i.e., the Mehler transform pair.*

*(ii) It is worth noting the close analogy between the non-Euclidean Fourier transform pair* (69) *and* (70) *and the Euclidean Fourier transform in the plane* R2*. The Fourier transform f of a function f on* , R<sup>2</sup> *is given by*

$$\tilde{f}(\mu\omega) = \int\_{\mathbb{R}^2} f(\mathbf{x}) e^{-i\mu \cdot (\mathbf{x}, \omega)} \,\mathrm{d}\mathbf{x}.$$

*Then, the Fourier inversion formula, valid, for example, if f* ∈ *C*<sup>∞</sup>*c* (R<sup>2</sup>)*, reads*

$$f(\mathfrak{x}) = \frac{1}{(2\pi)^2} \int\_{\mathbb{R}^+} \int\_{\mathbb{S}^1} \widetilde{f}(\mu \omega) \, e^{i\mu \cdot (\mathfrak{x}, \omega)} \, \mu \, \mathrm{d}\omega \, \mathrm{d}\mu \, \mathrm{d}\mu$$

d*ω denoting the circular measure on* S1 *(p. 4, [2]).*

3.4.3. Connection between the Spherical Laplace Transform at Re *λ* = −12 and the Non-Euclidean Fourier Transform

We can now prove the following theorem.

**Theorem 5.** *Suppose that the sequence* { *fn*}<sup>∞</sup>*n*=0*, where fn* = (*n* + 1)<sup>2</sup> *an (an being the Legendre coefficients of expansion* (14)*) satisfies the Hausdorff condition* (20)*. Then:*

*(i) The spherical Laplace transform at* Re *λ* = −12 *, restricted to the odd component (in μ) of* ,*a*(*σ* + <sup>i</sup>*μ*) *(see Section 3.1) reads*

$$\check{F}(\mu) = \int\_0^{+\infty} \underline{F}(\cosh \upsilon) P\_{-\frac{1}{2} + \mathrm{i}\mu}(\cosh \upsilon) \sinh \upsilon \, d\upsilon,\tag{72}$$

*where the equality holds in the sense of the L*2*-norm. Precisely, F*(cosh *v*) ∈ *L*<sup>2</sup>(1, <sup>+</sup>∞) *and <sup>F</sup>*,(*μ*) *is an even function of μ which belongs to <sup>L</sup>*<sup>2</sup>(R+, *μ* tanh *πμ* <sup>d</sup>*μ*)*.*

> *(ii) The inverse of formula* (72) *reads*

$$F(\upsilon) = \underline{F}(\cosh \upsilon) = \int\_0^{+\infty} \check{F}(\mu) P\_{-\frac{1}{2} + i\mu}(\cosh \upsilon) \,\mu \tanh \tau \mu \,d\mu. \tag{73}$$

*(iii) The following equality holds:*

$$\int\_0^{+\infty} \left| \widetilde{F}(\mu) \right|^2 \mu \tanh \pi \mu \, \mathrm{d}\mu = \int\_1^{+\infty} \left| \mathbb{E}(\cosh v) \right|^2 \, \mathrm{d}(\cosh v). \tag{74}$$

*(iv) The spherical Laplace transform coincides at* Re *λ* = −12*with the non-Euclidean Fourier transform.*

**Proof.** We start by proving statements (ii) and (iii) and, successively, we shall prove (i) and (iv).

(ii) In view of the assumptions on { *fn*}<sup>∞</sup>*n*=0, we may use the results of Theorem 3 and from formula (50) with *σ* ≡ Re *λ* = −12 :

$$F(\upsilon) = \underline{F}(\cosh \upsilon) = \frac{1}{4\pi} \int\_{-\infty}^{+\infty} \tilde{a}\left(-\frac{1}{2} + i\mu\right) (2i\mu) P\_{-\frac{1}{2} + i\mu}(\cosh \upsilon) \,\mathrm{d}\mu. \tag{75}$$

Let *F* , (*μ*) .= , *<sup>a</sup>*(−1/2+i*μ*) −i*π* tanh *πμ* , as we already did when passing from (53) to (54b). Formula (75) can be rewritten as 

$$F(\upsilon) = \underline{F}(\cosh \upsilon) = \int\_0^{+\infty} \widetilde{F}(\mu) \, P\_{-\frac{1}{2} + i\mu}(\cosh \upsilon) \, \mu \, \tanh \tau \mu \, \mathrm{d}\mu,\tag{76}$$

which coincides with the inverse spherical Laplace transform (73). Note that formula (76) coincides with the inversion of the non-Euclidean Fourier transform (70) in the specific case of radial functions.

(iii) First, we prove that the l.h.s. of (74) is convergent. The integral +∞ 0 |*F*,(*μ*)|<sup>2</sup>*μ* tanh *πμ* d*μ* is split into the integral over [0, 1] plus the integral over [1, <sup>+</sup>∞). We have 10 |*F*,(*μ*)|<sup>2</sup>*μ* tanh *πμ* d*μ* = 1*π*<sup>2</sup> 10 |,*a*(−12 + <sup>i</sup>*μ*)|<sup>2</sup> *μ* tanh *πμ* d*μ*, which is convergen<sup>t</sup> since lim*μ*→<sup>0</sup>(*μ*/ tanh *πμ*) = 1/*<sup>π</sup>*, and ,*<sup>a</sup>*(−12 + <sup>i</sup>*μ*) belongs to *L*<sup>2</sup>(0, 1) for inequality (25). The integral +∞ 1 |*F*,(*μ*)|<sup>2</sup>*μ* tanh *πμ* d*μ* is convergen<sup>t</sup> since |tanh *πμ*|−<sup>1</sup> *M* (*M* constant) for *μ* ∈ [1, <sup>+</sup>∞) and +∞ 1 |√*μ* ,*<sup>a</sup>*(−12 + <sup>i</sup>*μ*)|<sup>2</sup> d*μ* < +∞ 1 |*μ*2 ,*<sup>a</sup>*(−12 + <sup>i</sup>*μ*)|<sup>2</sup> d*μ* < ∞ in view of statement (ii) of Proposition 2. We have proved above that (76) coincides with the inverse non-Euclidean Fourier transform (70). From statement (ii) of Theorem 4, we know that the mapping *f* → *f* , (which, in our case, corresponds to the mapping *<sup>F</sup>*(*v*) = *F*(cosh *v*) → *F* , (*μ*)) is an isometry of *<sup>L</sup>*<sup>2</sup>(*<sup>D</sup>*, d*z*) onto *L*<sup>2</sup>(R<sup>+</sup> × *B*, 12*π μ* tanh *πμ* <sup>d</sup>*μ*d*b*). Therefore, the convergence of integral +∞ 0 |*F*,(*μ*)|<sup>2</sup>*μ* tanh *πμ* d*μ* along with the above mentioned isometry allows us to state equality (74) and that *F*(cosh *v*) ∈ *L*<sup>2</sup>(1, <sup>+</sup>∞).

(i) Consider formula (54b), which is the formal expression of the spherical Laplace transform at Re *λ* = −12 (restricted to its odd component with respect to the variable *μ* ∈ R). We note that it coincides with formula (72) which, by formula (74) proved above, holds as an equality in the sense of the *L*2-norm, as specified by statement (i).

(iv) We have indeed proved that formulas (72) and (73) coincide, respectively, with formulas (71a) and (71b), which are the Mehler reduction of the non-Euclidean Fourier transform pair (69) and (70).
