**4. Conclusions**

In summary, we have proved the following results. Let the Fourier–Legendre expansion 14*π* ∑∞*<sup>n</sup>*=<sup>0</sup>(<sup>2</sup>*<sup>n</sup>* + <sup>1</sup>)*anPn*(cos *<sup>θ</sup>*), and assume the numbers { *fn*}<sup>∞</sup>*n*=<sup>0</sup> (*fn* = (*n* + <sup>1</sup>)<sup>2</sup>*an*) to satisfy the Hausdorff-type condition (20). Then:

1. (a) The Fourier–Legendre expansion converges to a function *f*(cos *u*) analytic in the interval −1 < cos *u* < 1.

(b) The function *f*(cos *u*) admits a holomorphic extension to the complex (cos *<sup>θ</sup>*)-plane (*θ* = *u* + i*v*) cut along the semi-axis [1, <sup>+</sup>∞).

2. (a) The Fourier–Legendre coefficients {*an*}<sup>∞</sup>*n*=<sup>0</sup> are the restrictions to non-negative integers of a transform, called spherical Laplace transform, which is the composition of the ordinary Laplace transform with the Abel transform of *<sup>F</sup>*(*v*) ≡ *F*(cosh *<sup>v</sup>*), which is the jump function across the cut [1, <sup>+</sup>∞). Namely, for Re *λ* > −12 ,

$$\widetilde{a}(\lambda) = \int\_0^{+\infty} e^{-(\lambda + \frac{1}{2})w} \left( \mathcal{A}F \right)(w) \, \mathrm{d}w = 2 \int\_0^{+\infty} \underline{F}(\cosh \upsilon) \, Q\_{\lambda}(\cosh \upsilon) \sinh \upsilon \, \mathrm{d}\upsilon,\tag{77}$$

*Qλ*(·) being the Legendre function of the second kind.

(b) The function ,*<sup>a</sup>*(*λ*), holomorphic in the half-plane Re *λ* > −12 , satisfies Carlson's bound and interpolates uniquely the coefficients {*an*}<sup>∞</sup>*n*=0, that is, *an* = ,*<sup>a</sup>*(*λ*)|*<sup>λ</sup>*=*n*.

3. The jump function across the cut admits the following integral representation:

$$\begin{split} F(\upsilon) &= \underline{F}(\cosh \upsilon) = \frac{1}{2\pi} \int\_{-\infty}^{+\infty} \tilde{a}(\sigma + i\mu) \left(\sigma + \frac{1}{2} + i\mu\right) P\_{\upsilon + i\mu}(\cosh \upsilon) \, \mathrm{d}\mu \\ &= \frac{1}{2\pi i} \int\_{\sigma - i\mu}^{\sigma + i\infty} \tilde{a}(\lambda) \left(\lambda + \frac{1}{2}\right) P\_{\lambda}(\cosh \upsilon) \, \mathrm{d}\lambda \qquad \left(\sigma \rhd - \frac{1}{2}\right), \end{split} \tag{78}$$

where *<sup>P</sup>λ*(·) denotes the Legendre function of the first kind. Representation (78) is the inverse of the spherical Laplace transform (77).

4. (a) For Re *λ* = −12 the spherical Laplace transform, restricted to the odd component in *μ* of ,*a*(*σ* + <sup>i</sup>*μ*), reads 

$$\hat{F}(\mu) = \int\_0^{+\infty} \underline{F}(\cosh \upsilon) P\_{-\frac{1}{2} + i\mu}(\cosh \upsilon) \sinh \upsilon \, d\upsilon,\tag{79}$$

where *F* , (*μ*) = , *<sup>a</sup>*(O)(− 12 <sup>+</sup>i*μ*) −i*π* tanh *πμ* (,*a*(O)(−12 + <sup>i</sup>*μ*) denoting the odd component of ,*<sup>a</sup>*(−12 + <sup>i</sup>*μ*)). Equality (79) holds in the sense of *L*2-norm: *F*(cosh *v*) ∈ *L*<sup>2</sup>(1, <sup>+</sup>∞), *<sup>F</sup>*,(*μ*) ∈ *<sup>L</sup>*<sup>2</sup>(R+, *μ* tanh *πμ* <sup>d</sup>*μ*). The inverse of formula (79) reads

$$F(v) = \underline{F}(\cosh v) = \int\_0^{+\infty} \tilde{F}(\mu) P\_{-\frac{1}{2} + i\mu}(\cosh v) \, \mu \tanh \tau \mu \, \mathrm{d}\mu. \tag{80}$$

(b) Formulas (79) and (80) can be written explicitly, passing through Mehler transform, in terms of non-Euclidean Fourier transform as follows:

$$\widetilde{f}(\mu, b) = \int\_{D} e^{(\frac{1}{2} - \langle \mu \rangle) \langle z, b \rangle} f(z) \, \mathrm{d}z \qquad (\mu \in \mathbb{R}, b \in \mathcal{B}), \tag{81}$$

where *f* , (*μ*, *b*) = 2*πF* , (*μ*) and d*z* is the invariant surface element on the non-Euclidean disk *D*. The inverse of (81) is

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

where d*b* is the angular measure on the boundary *B* of the unit disk *D*.

(c) The functions *P*− 1 2<sup>+</sup>i*μ*(cosh *v*) can be represented as follows:

$$P\_{-\frac{1}{2}+i\mu}(\cosh \upsilon) = \frac{1}{2\pi} \int\_{B} e^{(\frac{1}{2}-i\mu)\langle z, b\rangle} \,\mathrm{d}b \qquad (\mu \in \mathbb{R}, z \in D, |z| = \tanh(\upsilon/2)),$$

where: *z*, *b* = ln <sup>P</sup>(*<sup>z</sup>*, *b*) is the *signed* non-Euclidean distance from the center of *D* to the horocycle with normal *b* passing through *z* ∈ *D*, and P denotes the Poisson kernel.

(d) The conical functions *P*− 1 2 <sup>+</sup>i*μ*(cosh *v*) = *P*− 12 −i*μ*(cosh *v*) correspond to the fundamental series of the unitary irreducible representation of the group SU(1, <sup>1</sup>), which acts transitively on the non-Euclidean disk *D*.

5. Last but not least, we wish to spend a few words to emphasize the differences between the classical Stein and Wainger approach and ours. First of all, we want to stress the grea<sup>t</sup> relevance of the pioneering work of Stein and Wainger; nevertheless, we believe that some remarks are in order.

(a) Stein and Wainger [9] assume that the Legendre coefficients are the restriction to the integers of a function (denoted by *a*(*s*) in their notation), which belongs to a space *<sup>H</sup>*2∗(Re *s* > −12 ). This latter is the space of functions *a*(*s*) which belong to *H*<sup>2</sup>(Re *s* > −12 ) and, for which, in addition, the squared norm *a*<sup>2</sup>∗ = +∞ −∞ |*a*(−12 + i*t*) − *<sup>a</sup>*(−12 − *it*)|<sup>2</sup> *t* d*t* tanh *πt* is finite. In their approach, it remained open and rather obscure the following question: How can it be established if the coefficients {*an*} are the restriction of a function belonging to *<sup>H</sup>*2∗(Re *s* > −12 )? Conversely, in our approach, we start directly from the Legendre coefficients, which are required to satisfy a suitable Hausdorff-type condition strictly connected with the moment theory. This second approach seems more direct, especially in the applications (e.g., scattering theory), where only the coefficients of the expansion are known.

(b) A geometrical analysis of the problem (see Section 3 of this paper) is missing in Stein and Wainger's work. Correspondingly, the remarkable results of Helgason on the non-Euclidean Fourier analysis are not mentioned.

(c) In Stein and Wainger's paper, the analytical properties of the Spherical Laplace Transform, as well as its character of being the composition of a Laplace and an Abel-type transform, do not emerge. This also makes the connection between the Mehler transform and the spherical Laplace transform not transparent.

**Funding:** This research received no external funding.

**Conflicts of Interest:** The author declares no conflict of interest.
