**Horocyclic Coordinates:**

We now introduce another system of coordinates (*<sup>τ</sup>*, *ζ*) as follows:

$$z\_0 = -\mathbf{i}\sin\tau + \frac{1}{2}\zeta^2 e^{-i\tau},\tag{4a}$$

$$z\_1 = \zeta e^{-\iota u}, \tag{4b} \qquad\qquad\qquad (\tau, \zeta \in \mathbb{C}), \tag{4b}$$

$$z\_2 = \cos \tau - \frac{1}{2} \zeta^2 e^{-\text{tr}}.\tag{4c}$$

For *τ* = i*w* (*w* ∈ R) and *ζ* ∈ R, Equation (4) reads

$$\mathbf{x}\_0 = \sinh w + \frac{1}{2}\zeta^2 e^w,\tag{5a}$$

$$\mathbf{x}\_1 = \zeta\_\circ e^w,\tag{5b} \in \mathbb{R}^w \tag{5c} \\ \tag{5b}$$

$$\text{tr}2 = \cosh w - \frac{1}{2}\zeta^2 \,\text{e}^w.\tag{5c}$$

.

Then, we consider the intersection points *xw* of *X* -+ with the family of planes *Pw* : *x*0 + *x*2 = *e<sup>w</sup>* (*w* ∈ R) (see Equation (5)), i.e., the points *xw* = (sinh *w*, 0, cosh *<sup>w</sup>*). The sections of *X*2 by these planes are the (real) parabolae <sup>Π</sup>+*w* (except in the case *x*0 + *x*2 = 0). When *ζ* = 0 in (5), we obtain the point *xw*, which is the apex of the corresponding parabola <sup>Π</sup>+*w*(see Figure 1).

This geometrical construction can now be extended to the complex one-sheeted hyperboloid *X*(*c*) 2 We consider the (complex) hyperbola *X* -(*c*) lying in the plane *z*1 = 0, whose equation is: *z*20 − *z*22 = −1 (see Equation (4)), and its intersections with the family of planes *Pτ* with equation *z*0 + *z*2 = *e*<sup>−</sup>i*<sup>τ</sup>* (*τ* ∈ C ˙ . = C/2*π*Z). Each plane *Pτ* intersects the hyperbola *X* -(*c*) at the (unique) point *zτ* = (*<sup>z</sup>*0 = −i sin *τ*, *z*1 = 0, *z*2 = cos *<sup>τ</sup>*), thus defining a bijection from the set of planes P .= {*<sup>P</sup>τ* : *τ* ∈ C ˙ } onto *X* -(*c*) = {*<sup>z</sup>τ* : *τ* ∈ C˙ }. For each *τ* ∈ C˙ , the manifold *Pτ* ∩ *X*(*c*) 2 is a complex parabola Π*<sup>τ</sup>*, which we call a *complex horocycle*. The set of horocycles {<sup>Π</sup>*τ* : *τ* ∈ C ˙ } defines a fibration with basis *X* -(*c*) on the dense domain *<sup>X</sup>*2(*c*) .= {*z* ∈ *X*(*c*) 2 : *z*0 + *z*2 = 0} of *X*(*c*) 2 . We can therefore associate with this fibration the parametric representation of *<sup>X</sup>*2(*c*) given by Equation (4), (*<sup>τ</sup>*, *ζ*) being the horocyclic coordinates of the point *z* ∈ *<sup>X</sup>*2(*c*).

### *2.2. The Radon Transform*

We now define a Radon-type transformation in *X*2, where the horocycles defined above play the same role as the planes do in the ordinary Radon transformation. We introduce the following integral:

$$\int\_{h\_w} \underline{F} \left( \cosh w - \frac{1}{2} \zeta^2 \ e^w \right) \, \mathrm{d}\zeta = \hat{F}(w), \tag{6}$$

where *hw* is the oriented segmen<sup>t</sup> of horocycle, which is represented by the arc of parabola whose apex is obtained by setting *ζ* = 0 in (5) (i.e., its coordinates are: *x*0 = sinh *w*, *x*1 = 0, *x*2 = cosh *w*; *w* ∈ R), and whose endpoints lie on the plane *x*2 = 1. The function *F* is assumed to satisfy the regularity conditions that make integral (6) convergent.

**Remark 1.** *Legendre expansions, which will be our concern later, involve functions depending only on* cos *u* = *x*2 *(see* (2c)*). This is why, in the integral* (6)*, we limit ourselves to consider functions only of the form F* = *F*(cosh *w* − 1 2 *ζ*<sup>2</sup>*e<sup>w</sup>*) *(see* (5c)*).*

Since the integrand in (6) is an even function of *ζ*, the integration domain can be restricted to the part *h*<sup>+</sup> *w* of *hw* with *x*1 0, which can be parametrized as follows (see (5)):

$$h\_w^+ = \{ \mathbf{x} \in \Pi\_w^+; \mathbf{x} = \mathbf{x}(\zeta, w); \zeta(\delta) = \left[ 2e^{-w}(1 - \delta)(\cosh w - 1) \right]^{\frac{1}{2}}; 0 \le \delta \ll 1 \}\tag{7} \qquad (w \in \mathbb{R}),\tag{7}$$

positively oriented from the apex to the endpoint lying on *x*2 = 1. For *δ* = 1, we have *ζ* = 0, which yields the apex of the parabola representing the horocycle; for *δ* = 0, we have *ζ* = [<sup>2</sup>*e*<sup>−</sup>*<sup>w</sup>*(cosh *w* − 1)]1/2, which gives the intersection of the horocycle with the plane *x*2 = 1.

We may now introduce another parametrization of the segmen<sup>t</sup> of horocycle *h*<sup>+</sup> *w* , which is obtained by setting *x*2 = cosh *v* (see (3c)):

$$h\_w^+ = \{ \mathbf{x} \in \Pi\_w^+; \mathbf{x} = \mathbf{x}(\zeta, w); \zeta(v) = \left[ 2e^{-w} (\cosh w - \cosh v) \right]^{\frac{1}{2}}; 0 \le v \le w \} \tag{8} \\ \tag{8}$$

Indeed, we have *ζ* = 0 for *v* = *w* (the apex of the parabola) and *ζ* = [<sup>2</sup>*e*<sup>−</sup>*<sup>w</sup>*(cosh *w* − 1)]1/2 for *v* = 0 (endpoint of the parabola: *x*2 = cosh *v* = 1). Since d*ζ*/d*v* = −*e*<sup>−</sup>*<sup>w</sup>*/2 sinh *v*[2(cosh *w* − cosh *v*)]−1/2, integral (6) takes the form:

$$\hat{F}(w) = 2e^{-w/2} \int\_0^w \underline{F}(\cosh v) \frac{\sinh v}{[2(\cosh w - \cosh v)]^{\frac{1}{2}}} \,\mathrm{d}v \doteq e^{-w/2} \left(\mathcal{A}F\right)(w) \qquad (w \in \mathbb{R}), \tag{9}$$

where (A*F*)(*w*) is an Abel-type integral.

The fibration realized by the horocycles Π<sup>+</sup> *w* can now be extended to the complex domain by using the complex horocycles Π*τ* (*τ* = *t* + i*w*; *t*, *w* ∈ R), whose intersection with the (complex) meridian hyperbola *X*-(*c*) is the point *zτ* = ( −i sin *τ*, 0, cos *<sup>τ</sup>*). Accordingly, we introduce the following integral:

$$2\int\_{h\_{\tau}^{+}} \underline{f}\left(\cos\tau - \frac{1}{2}\zeta^{2}e^{-l\tau}\right)d\zeta = \widehat{f}(\tau),\tag{10}$$

*h*<sup>+</sup> *τ*being the (oriented) arc of the complex horocycle defined by (see (4) and (7)):

$$h\_{\tau}^{+} = \{ z \in \Pi\_{\tau}; z = z(\check{\zeta}, \tau); \check{\zeta}(\delta) = [2\check{\epsilon}^{1\tau}(1-\delta)(\cos \tau - 1)]^{\frac{1}{2}}; 0 \ll \delta \ll 1\} \qquad (\tau \in \mathbb{C}).$$

For *δ* = 1, we have *ζ* = 0, i.e., the point *zτ* belonging to *<sup>X</sup>*-(*c*), while for *δ* = 0 we have *ζ* = [<sup>2</sup>*e*i*τ*(cos *τ* − 1)]1/2, which is the intersection of *h*<sup>+</sup> *τ* with the plane *z*2 = 1. Similarly to what done before, we may now introduce the following parametrization of *h*<sup>+</sup> *τ* , obtained by setting *z*2 = cos *θ* (*θ* ∈ C, *θ* = *u* + i*v*; see (1) and (8)):

$$h\_{\tau}^{+} = \{ z \in \Pi\_{\tau}; z = z(\zeta, \tau); \zeta(\theta) = [2\ell^{\mathrm{i}\tau}(\cos \tau - \cos \theta)]^{\frac{1}{2}}; \theta \in \gamma\_{\tau} \} \qquad (\tau \in \mathbb{C}),$$

where *γτ* denotes the ray in the *θ*-plane oriented from 0 to *τ*:

$$\gamma\_{\tau} = \{ \theta = \theta(\delta); \cos \theta(\delta) - 1 = \delta(\cos \tau - 1); 0 \le \delta \ll 1 \} \qquad (\tau \in \mathbb{C}).$$

> Now, since d*ζ*/d*θ* = *e*i*τ*/2[2(cos *τ* − cos *<sup>θ</sup>*)]−1/2 sin *θ*, integral (10) can be rewritten in the form:

$$\hat{f}(\tau) = -2e^{j\tau/2} \int\_{\gamma\_{\tau}} f(\theta) \frac{\sin \theta}{[2(\cos \tau - \cos \theta)]^{\frac{1}{2}}} \, d\theta \,\tag{11}$$

where *f*(*θ*) ≡ *f*(cos *<sup>θ</sup>*). The relevant branch of the function [2(cos *τ* − cos *<sup>θ</sup>*)]−1/2 is specified by the condition that, for *τ* = i*w* and *θ* = i*v* (with *w* > *v*), it takes the value [2(cosh *w* − cosh *v*)]−1/2 0. Putting in (11) *τ* = i*w*, *θ* = i*v*, we re-obtain precisely the r.h.s. of formula (9), once *f* is identified with *F*.

Restricting formula (11) to the set of real values for the variables *τ* and *θ*, namely, *τ* = *t* and *θ* = *u* (*t*, *u* ∈ R), from (11), we obtain

$$\widehat{f}(t) = -2\varepsilon^{\mathrm{i}t/2} \int\_0^t f(u) \frac{\sin u}{[2(\cos t - \cos u)]^{\frac{1}{2}}} \,\mathrm{d}u. \tag{12}$$

Accounting for the relevant branch of the factor [2(cos *τ* − cos *<sup>θ</sup>*)]−1/2 in (11), formula (12) can be written in the following more precise form (involving a positive bracket):

$$\hat{f}(t) = -2\text{i}\,\varepsilon(t)\,\varepsilon^{\text{i}t/2} \int\_{0}^{t} f(u) \, \frac{\sin u}{[2(\cos u - \cos t)]^{\frac{1}{2}}} \,\text{d}u,\tag{13}$$

where *ε*(*t*) denotes the sign function.

*2.3. Holomorphic Extension Associated with Trigonometric Series*

2.3.1. Fourier–Legendre Expansions as Trigonometric Series

Consider the following Legendre series

$$\frac{1}{4\pi} \sum\_{n=0}^{\infty} \left(2n+1\right) a\_{\text{ll}} P\_{\text{ll}}(\cos u),\tag{14}$$

where *Pn*(·) denotes the Legendre polynomials, which satisfy the following integral representation:

$$P\_n(\cos u) = \frac{1}{\pi} \int\_0^\pi (\cos u + \mathbf{i} \sin u \cos \eta)^n \, \mathrm{d}\eta \dots$$

Suppose that expansion (14) converges to a function *f*(cos *u*) absolutely integrable in the interval *u* ∈ [0, *<sup>π</sup>*]. Then, the Legendre coefficients *an* can be written as

$$a\_{ll} = 2\pi \int\_0^{\pi} \underline{f}(\cos u) \, P\_n(\cos u) \, \sin u \, \mathrm{d}u.$$

In Ref. [21], we proved the following proposition.

**Proposition 1.** *The Legendre coefficients* {*an*}<sup>∞</sup>*n*=<sup>0</sup> *coincide with the Fourier coefficients of the following form:*

$$a\_n = \int\_{-\pi}^{\pi} \hat{f}(t) \, e^{\text{int}} \, dt \qquad (n = 0, 1, 2, \dots), \tag{15}$$

*where*

$$\hat{f}(t) = -2\operatorname{i}\varepsilon(t)\,\varepsilon^{\text{l}t/2} \int\_{0}^{t} f(u) \, \frac{\sin u}{[2(\cos u - \cos t)]^{\frac{1}{2}}} \, \text{d}u,\tag{16}$$

*with f*(*u*) ≡ *f*(cos *<sup>u</sup>*)*, and ε*(*t*) *being the sign function.*

**Proof.** See Proposition 3.1 of Ref. [21].

> Note that (16) coincides with (13). From (16), it is easy to verify that

$$
\widehat{f}(t) = -\epsilon^{\text{it}} \widehat{f}(-t),
\tag{17}
$$

which, through (15), yields the following symmetry relation for the Legendre coefficients:

$$a\_n = -a\_{-n-1} \qquad (n \in \mathbb{Z}).\tag{18}$$

Now, we can introduce the following trigonometric series, assuming (18) to hold:

$$\begin{split} \frac{1}{2\pi} \sum\_{n=-\infty}^{+\infty} a\_n e^{-\text{int}} &= \frac{1}{2\pi} \left[ \sum\_{n=0}^{+\infty} a\_n e^{-\text{int}} - e^{\text{it}} \sum\_{n=0}^{+\infty} a\_n e^{\text{int}} \right] \\ &= \frac{1}{2\pi} e^{\text{i}(t-\pi)/2} \sum\_{n=-\infty}^{+\infty} (-1)^n a\_n \cos\left[ \left( n + \frac{1}{2} \right) (t-\pi) \right] = \frac{1}{2\pi} e^{\text{i}(t-\pi)/2} \sum\_{n=-\infty}^{+\infty} a\_n \sin\left[ \left( n + \frac{1}{2} \right) t \right], \end{split} \tag{19}$$

and study the associated holomorphic extension.

2.3.2. Holomorphic Extension Associated with Trigonometric Series

Consider a sequence { *fn*}∞ *<sup>n</sup>*=0 of (real) numbers, and denote by Δ the difference operator:

$$
\Delta f\_n = f\_{n+1} - f\_n.
$$

We have:

$$\Delta^k f\_n = \underbrace{\Delta \times \Delta \times \cdots \times \Delta}\_{k-\text{times}} f\_n = \sum\_{m=0}^k (-1)^m \binom{k}{m} f\_{n+k-m}$$

(for any integer *k* 0); Δ<sup>0</sup> is the identity operator, by definition. Suppose that there exists a positive constant *M* such that

$$(n+1)^{\left(1+\varepsilon\right)} \sum\_{i=0}^{n} \binom{n}{i}^{\left(2+\varepsilon\right)} \left| \Delta^i f\_{\left(n-i\right)} \right|^{\left(2+\varepsilon\right)} < M \qquad (n = 0, 1, 2, \dots; \varepsilon > 0). \tag{20}$$

We shall refer to (20) as the *Hausdorff condition* for its relevance in the solution of the Hausdorff moment problem [8]. The tool we use to guarantee uniqueness of the interpolation of a sequence of numbers { *fn*}∞ *<sup>n</sup>*=0 is Carlson's theorem [22]. Essentially, it gives growth conditions under which a function is uniquely determined by its values on non-negative integers. Let us recall that an entire function *f*(*z*) is of exponential type *τ* < ∞ if

$$\limsup\_{r \to \infty} \frac{M\_f(r)}{r} = \tau,\tag{21}$$

where *Mf*(*r*) denotes the maximum modulus of *f*(*z*) for |*z*| = *r*. The rate of growth of entire functions can be specified along different directions by the Phragmén-Lindelöf indicator function,

$$h\_f(\theta) = \limsup\_{r \to \infty} \frac{\log \left| f(re^{i\theta}) \right|}{r}. \tag{22}$$

Note that the preceding definitions can be extended to functions which are not entire, but regular (that is, analytic and single-valued) in a sector with vertex at the origin [22].

**Theorem 1** (Carlson's theorem (Section 9.2, p. 153, [22]))**.** *Let f*(*z*) *be regular in the half-plane* Re *z* 0 *and*

*(i) f*(*z*) *is of exponential type τ* < <sup>∞</sup>*, (ii) hf*(*π*/2) + *hf*(− *π*/2) < 2*π, (iii) f*(*n*) = 0 *for n* = 0, 1, 2, . . .*,*

*then f*(*z*) *vanishes identically.*

Among the functions of exponential type, condition (ii) requires *f*(*z*) to be of exponential type less than *π* on the imaginary axis that is: *f*(<sup>i</sup>*y*) = *O*(1) exp(*c*|*y*|) for some *c* < *π*. We shall refer to conditions (i) and (ii) as *Carlson's bound*. Moreover, an analytic function which interpolates a sequence of numbers { *fn*}∞ *<sup>n</sup>*=0 and satisfy conditions (i) and (ii) above will be called a *Carlsonian interpolant*, in view of the fact that the uniqueness of the interpolation is guaranteed by Carlson's theorem.

**Proposition 2.** *Suppose that the set of numbers* { *fn*}∞ *<sup>n</sup>*=0*, with fn* .= (*n* + <sup>1</sup>)<sup>2</sup>*an (an being the Fourier–Legendre coefficients of the series* (14)*), satisfies condition* (20)*. Then:*

*(i) There exists a unique Carlsonian interpolant* ,*a*(*λ*) *(λ* ∈ C,Re *λ* −1 2 *) of the coefficients* {*an*}<sup>∞</sup> *<sup>n</sup>*=0*, which is holomorphic in the half-plane* Re *λ* > −1 2*. Moreover,* ,*a*(*λ*) *belongs to the Hardy space H*<sup>2</sup>(C(+) −1/2)*.*

*.*


**Proof.** (i) Since the sequence { *fn*}∞ *<sup>n</sup>*=0 satisfies condition (20), the numbers *fn* are moments of a suitable function, that is, the following representation holds [8]:

$$f\_n = \int\_0^1 \mathbf{x}^n \,\varphi(\mathbf{x}) \, \mathrm{d}\mathbf{x} \qquad (n = 0, 1, 2, \dots), \tag{23}$$

 *.*

where *ϕ* ∈ *L*(<sup>2</sup>+*ε*)(0, <sup>1</sup>). Let *x* = *e*<sup>−</sup>*<sup>s</sup>* in formula (23):

$$f\_n = \int\_0^{+\infty} \varepsilon^{-(n+\frac{1}{2})s} \varepsilon^{-s/2} \,\, \rho(\varepsilon^{-s}) \,\, \text{d}s \qquad (n = 0, 1, 2, \dots).$$

Then, the numbers { *fn*}∞ *<sup>n</sup>*=0 can be formally regarded as the restriction to the (non-negative) integers of the following Laplace transform:

$$\widetilde{f}(\lambda) = \int\_0^{+\infty} e^{-(\lambda + \frac{1}{2})s} e^{-s/2} \,\varphi(e^{-s}) \,\mathrm{d}s \qquad \left(\mathrm{Re}\,\lambda > -\frac{1}{2}\right),\tag{24}$$

being *f* ,(*λ*)|*<sup>λ</sup>*=*<sup>n</sup>* = *fn*. We see that *e*<sup>−</sup>*<sup>s</sup>*/2*ϕ*(*e*<sup>−</sup>*<sup>s</sup>*) ∈ *L*<sup>2</sup>(0, + ∞): in fact, +∞ 0 |*e*<sup>−</sup>*<sup>s</sup>*/2*ϕ*(*e*<sup>−</sup>*<sup>s</sup>*)| 2 d*s* = 1 0 |*ϕ*(*x*)| 2 d*x* < <sup>∞</sup>, since *ϕ* ∈ *L*(<sup>2</sup>+*ε*)(0, 1) and, a fortiori, *ϕ*(*x*) ∈ *L*<sup>2</sup>(0, <sup>1</sup>). Then, by the Paley–Wiener theorem [23], we have *f* ,(*λ*) ∈ *H*<sup>2</sup>(C(+) −1/2), which is the Hardy space whose norm is: *f* ,2 . = sup*σ*>−1/2( +∞ −∞ | *f* ,(*σ* + <sup>i</sup>*μ*)| 2 <sup>d</sup>*μ*)1/2, and C(+) −1/2 . = {*λ* ∈ C, *λ* = *σ* + i*μ*; *σ* > −1 2 , *μ* ∈ <sup>R</sup>}. Hence, use can

be made of Carlson's theorem, which guarantees that *f* (*λ*) is the unique Carlsonian interpolant of the sequence { *fn*}∞ *<sup>n</sup>*=0. Let ,*a*(*λ*) . = *f* ,(*λ*)/(*λ* + 1)2. ,*a*(*λ*) is a function holomorphic for Re *λ* > −1 2 , which satisfies Carlson's bound since *f* ,(*λ*) does. Then, ,*a*(*λ*) is the unique Carlsonian interpolant of the set of Fourier–Legendre coefficients {*an*}<sup>∞</sup> *<sup>n</sup>*=0 since ,*<sup>a</sup>*(*λ*)|*<sup>λ</sup>*=*<sup>n</sup>* = *f* ,(*λ*) (*<sup>λ</sup>*+<sup>1</sup>)<sup>2</sup> *λ*=*<sup>n</sup>* = *fn* (*n*+<sup>1</sup>)<sup>2</sup> = *an*. Finally, it is readily shown that ,*a*(*λ*) ∈ *H*<sup>2</sup>(C(+) −1/2):

$$\|\hat{a}\|\_{2}^{2} = \sup\_{\sigma > -\frac{1}{2}} \int\_{-\infty}^{+\infty} \left| \frac{\hat{f}(\sigma + i\mu)}{(\sigma + 1 + i\mu)^{2}} \right|^{2} \, \mathrm{d}\mu \leqslant 16 \sup\_{\sigma > -\frac{1}{2}} \int\_{-\infty}^{+\infty} \left| \hat{f}(\sigma + i\mu) \right|^{2} \, \mathrm{d}\mu < \infty \,\tag{25}$$

,

and statement (i) is then proved. Regarding point (ii), since *f* ,(*λ*) ∈ *H*<sup>2</sup>(C(+) −1/2), then *f* ,(*σ* + <sup>i</sup>*μ*) ∈ *<sup>L</sup>*<sup>2</sup>(− <sup>∞</sup>, + ∞) for any fixed value of *σ* −1 2 . Regarding point (iii), since *f* ,(*λ*) ∈ *H*<sup>2</sup>(C(+) −1/2), then *f* ,(*λ*) tends to zero as *λ* → ∞ inside any fixed half-plane Re *λ δ* > −1 2 [23]. By Schwarz's inequality, we have for *σ* −1 2 ,

$$\int\_{-\infty}^{+\infty} |(\boldsymbol{\sigma} + \mathbf{i}\mu) \cdot \widetilde{\mathfrak{a}}(\boldsymbol{\sigma} + \mathbf{i}\mu)| \, \mathrm{d}\mu \leqslant \left( \int\_{-\infty}^{+\infty} \left| \frac{\boldsymbol{\sigma} + \mathbf{i}\mu}{(\boldsymbol{\sigma} + \mathbf{1} + \mathbf{i}\mu)^2} \right|^2 \, \mathrm{d}\mu \right)^{\frac{1}{2}} \Big( \int\_{-\infty}^{+\infty} \left| \widetilde{f}(\boldsymbol{\sigma} + \mathbf{i}\mu) \right|^2 \, \mathrm{d}\mu \Big)^{\frac{1}{2}} < \infty,$$

which proves statement (iv). Concerning point (v), return to the Laplace integral representation of *f* ,(*λ*) in (24). First, we want to prove that *e*<sup>−</sup>*<sup>s</sup>*/2*ϕ*(*e*<sup>−</sup>*<sup>s</sup>*) ∈ *L*<sup>1</sup>(0, + <sup>∞</sup>), which amounts to showing that 1 0 *ϕ*(*x*) √*x* d*x* < ∞. By Hölder's inequality,

$$\int\_0^1 \frac{|\!/\!\!\!\!\!/(\mathbf{x})|}{\sqrt{\mathbf{x}}} \, \mathrm{d}x \leqslant \left(\int\_0^1 |\!/\!\!\!/(\mathbf{x})|^{(2+\varepsilon)} \, \mathrm{d}x\right)^{\frac{1}{2+\varepsilon}} \cdot \left(\int\_0^1 \mathrm{x}^{-\frac{2+\varepsilon}{2+2\varepsilon}} \, \mathrm{d}x\right)^{\frac{1+\varepsilon}{2+\varepsilon}} < \infty,$$

where the rightmost integral converges since 2+*ε* 2+2*ε* < 1 for *ε* > 0, and *ϕ* ∈ *L*(<sup>2</sup>+*ε*)(0, <sup>1</sup>). Then, from representation (24):

$$\left(\left(\frac{1}{2} + \mathrm{i}\mu\right)^2 \widetilde{\mathfrak{a}}\left(-\frac{1}{2} + \mathrm{i}\mu\right) = \widetilde{f}\left(-\frac{1}{2} + \mathrm{i}\mu\right) = \int\_0^{+\infty} e^{-\mathrm{i}\mu s} e^{-s/2} \varrho(e^{-s}) \,\mathrm{d}s = \mathcal{F}\left\{ h(s) e^{-s/2} \varrho(e^{-s}) \right\} \,\mathrm{d}s$$

where F denotes the Fourier integral operator, and *h*(*s*) is the Heaviside step function. The Riemann–Lebesgue theorem guarantees that ,*<sup>a</sup>*(−<sup>1</sup> 2 + <sup>i</sup>*μ*) is a continuous function tending to zero as *μ* → ± <sup>∞</sup>, and statement (v) is proved. Finally, in order to prove statement (vi), we note that the Laplace transform (24) holds also for Re *λ* = −1 2 since *e*<sup>−</sup>*<sup>s</sup>*/2*ϕ*(*e*<sup>−</sup>*<sup>s</sup>*) ∈ *L*<sup>1</sup>(0, + ∞) ∩ *L*<sup>2</sup>(0, + <sup>∞</sup>). It follows that sup*σ*>−1/2 *μ*∈R | *f* ,(*σ* + <sup>i</sup>*μ*)| = | *f* , −1 2 + i*μ* |. Therefore, recalling that ,*a*(*λ*) = *f* ,(*λ*)/(*λ* + 1)2:

$$\sup\_{\substack{\sigma>-1/2\\\mu\in\mathbb{R}}} \left| \tilde{a}(\sigma+\mathrm{i}\mu) \right| = \sup\_{\substack{\sigma>-1/2\\\mu\in\mathbb{R}}} \frac{|\bar{f}(\sigma+\mathrm{i}\mu)|}{|\sigma+1+\mathrm{i}\mu|^2} = \frac{|\tilde{f}(-\frac{1}{2}+\mathrm{i}\mu)|}{\mu^2+1/4} = \left| \tilde{a} \left( -\frac{1}{2}+\mathrm{i}\mu \right) \right|.$$

,

where ,*<sup>a</sup>*(−<sup>1</sup> 2 + <sup>i</sup>*μ*) ∈ *<sup>L</sup>*<sup>1</sup>(− <sup>∞</sup>, + ∞) (see statements (iv) and (v)).

Let *ξ*0 0. We introduce in the complex plane C of the variable *τ* = *t* + i*w* (*t*, *w* ∈ R) the following domains: *τ*I(±*ξ*0) + . = {*τ* ∈ C : Im *τ* > <sup>±</sup>*ξ*0} and *τ*I(±*ξ*0) − . = {*τ* ∈ C : Im *τ* < <sup>±</sup>*ξ*0}. Correspondingly, we introduce the following cut-domains: *τ*I(*ξ*0) + \ *τ*Ξ(*ξ*0) + , where *τ*Ξ(*ξ*0) + . = {*τ* ∈ C : *τ* = 2*kπ* + i*w*, *w* > *ξ*0, *k* ∈

Z} (see Figure 2a) and *τ*I(−*ξ*0) − \ *τ*Ξ(−*ξ*0) − , where *τ*Ξ(−*ξ*0) − .= {*τ* ∈ C : *τ* = 2*kπ* + i*w*, *w* < −*ξ*0, *k* ∈ Z} (see Figure 2b). Finally, we denote by *A* ˙ . = *A*/2*π*Z any subset *A* of C, which is invariant under the translation group 2*π*Z. We are now ready to state the following proposition.

**Figure 2.** In the complex *τ*-plane, the grey regions represent: (**a**) the cut-domain *τ*I(*ξ*0) + \ *τ*Ξ(*ξ*0) + , (**b**) the cut-domain *τ*I(−*ξ*0) −\ *τ*Ξ(−*ξ*0) −. The cuts (thick lines) are located at Re *τ* = 2*k<sup>π</sup>*, *k* ∈ Z.

**Proposition 3.** *Consider the trigonometric series*

$$\frac{1}{2\pi} \sum\_{n=0}^{+\infty} a\_n \,\text{e}^{-in\tau} \qquad (\tau = t + iw; t, w \in \mathbb{R}),\tag{26}$$

*and suppose that the set of numbers* { *fn*}<sup>∞</sup>*n*=0*, with fn* .= (*n* + 1)<sup>2</sup> *an, satisfies condition* (20)*. Then:*


$$\left|\hat{F}^{(+)}(w)\right| \leqslant \|\hat{a}\_{\sigma}\|\_{1} \; \epsilon^{\sigma w} \qquad \left(\sigma \geqslant -\frac{1}{2}, w \in \mathbb{R}^{+}\right),$$

*where* ,*a*(*σ* + <sup>i</sup>*μ*) *(μ* ∈ R*) is the unique Carlsonian interpolant of the coefficients an, and*

$$\|\|\tilde{a}\_{\sigma}\|\|\_{1} \doteq \frac{1}{2\pi} \int\_{-\infty}^{+\infty} |\tilde{a}(\sigma + i\mu)| \,\mathrm{d}\mu < \infty \qquad \left(\sigma \rhd - \frac{1}{2}\right). \tag{27}$$


$$
\widetilde{a}(\sigma + i\mu) = \int\_0^{+\infty} \widehat{F}^{(+)}(w) \, e^{-(\sigma + i\mu)w} \, dw \qquad \left(\sigma > -\frac{1}{2}\right), \tag{28}
$$

.

*holomorphic in the half-plane σ* > −12 *. (vi) The following Plancherel equality holds:*

$$\int\_{-\infty}^{+\infty} |\widetilde{a}(\sigma + i\mu)|^2 \, \mathrm{d}\mu = 2\pi \int\_{-\infty}^{+\infty} \left| \mathring{F}^{(+)}(w) e^{-\sigma w} \right|^2 \, \mathrm{d}w \qquad \left(\sigma \gtrless -\frac{1}{2}\right).$$

**Proof.** Since the sequence { *fn*}<sup>∞</sup>*n*=<sup>0</sup> satisfies condition (20), then, given an arbitrary constant *C*, there exists an integer *n*0 such that |*an*| *C* for *n* > *n*0 and, accordingly,

$$\left| \frac{1}{2\pi} \sum\_{n=0}^{\infty} a\_n e^{-in\tau} \right| \leqslant \frac{\mathbb{C}}{2\pi} \sum\_{n=n\_0+1}^{\infty} e^{n\upsilon} \qquad \left( w \doteq \text{Im } \tau, \ \mathbb{C} = \text{constant} \right). \tag{29}$$

The series on the r.h.s. of (29) converges uniformly on any compact subdomain contained in the half-plane *w* < 0. Since

$$\frac{1}{2\pi} \sum\_{n=0}^{\infty} a\_{\text{ll}} \, e^{-in\tau} = \frac{1}{2\pi} \sum\_{n=n\_0+1}^{\infty} a\_{\text{ll}} \, e^{-in\tau} + T\_{\text{H}\_0}(\tau),$$

where *Tn*0 (*τ*) is a trigonometric polynomial, by the Weierstrass theorem on the uniformly convergen<sup>t</sup> series of analytic functions, series (26) converges uniformly on any compact subdomain of *τ*I(0) − to a function *f* -(+)(*τ*) holomorphic in *τ*I(0) − . Furthermore, since (*n* + 1)<sup>2</sup> *an* −−−→*n*→∞ 0, given an arbitrary constant *C*, there exists an integer *n*1 such that for *w* = 0:

$$\left| \frac{1}{2\pi} \sum\_{n=n\_1}^{\infty} a\_{\mathcal{U}} \, e^{-\text{inf}} \right| \leqslant \frac{1}{2\pi} \sum\_{n=n\_1}^{\infty} |a\_{\mathcal{U}}| \leqslant \frac{\mathcal{C}'}{2\pi} \sum\_{n=n\_1}^{\infty} \frac{1}{n^2} \leqslant \mathcal{C}' \frac{\pi}{12}.$$

Then, applying once again the Weierstrass theorem on the uniformly convergen<sup>t</sup> series of continuous functions, the series 12*π* ∑∞*<sup>n</sup>*=<sup>0</sup> *an e*<sup>−</sup>i*nt* converges to a continuous function *f*-(+)(*t*), and statement (i) is proved. Regarding statement (ii), we write the following integral:

$$\hat{f}\_{\eta}^{(+)}(t) = \frac{\mathrm{i}}{4\pi} \int\_{\mathcal{C}} \widetilde{a}(\lambda) \frac{e^{-i\lambda(t-\eta\,\pi)}}{\sin\pi\lambda} d\lambda \qquad (\eta = \pm), \tag{30}$$

where ,*a*(*λ*) (Re *λ* −12 ) is the unique Carlsonian interpolant of the coefficients {*an*}<sup>∞</sup>*n*=0, which is holomorphic in the half-plane Re *λ* > −12 (see statement (i) of Proposition 2). The contour C is contained in the half-plane C(+) −1/2, which encircles the semi-axis Re *λ* > −12 , and is chosen to cross the latter at a point *σ* > −12 , *σ* ∈ N (see Figure 3). Consider now the term exp[−i*<sup>λ</sup>*(*<sup>t</sup>*−*ηπ*)] sin *πλ* ; the following inequalities hold (*λ* = *σ* + <sup>i</sup>*μ*):

$$\left| \varepsilon^{-i(\sigma + i\mu)(t - \eta \pi)} \right| \leqslant 2 \cosh \pi \mu \quad \text{for} \quad \begin{cases} \quad 0 \leqslant t \leqslant 2\pi & \text{if} \quad \eta = +, \\\quad -2\pi \leqslant t \leqslant 0 & \text{if} \quad \eta = -, \end{cases} \tag{31}$$

$$\left|\sin\pi(\sigma+i\mu)\right|\geqslant\sinh\pi\mu,\tag{32}$$

$$|\sin\pi(\sigma+i\mu)| \geqslant |\sin\pi\sigma| \cosh\pi\mu. \tag{33}$$

From (31) and (32):

$$\left| \frac{e^{-i(\sigma + i\mu)(t - \eta \cdot \pi)}}{\sin \pi(\sigma + i\mu)} \right| \ll 2 \left| \frac{\cosh \pi \mu}{\sinh \pi \mu} \right| \quad \text{for} \quad \begin{cases} \quad 0 \leqslant t \leqslant 2\pi & \text{if} \quad \eta = +, \\\quad -2\pi \leqslant t \leqslant 0 & \text{if} \quad \eta = -, \end{cases} \tag{34}$$

while, combining (31) and (33):

$$\left|\frac{\varepsilon^{-i(\sigma+i\mu)(t-\eta\pi)}}{\sin\pi(\sigma+i\mu)}\right| \lessapprox \frac{2}{|\sin\pi\sigma|} < \infty \quad \text{for} \begin{cases} \quad 0 \lessleqslant t \lessapprox 2\pi & \text{if} \quad \eta = +,\\ \quad -2\pi \lessgtr t \lessapprox 0 & \text{if} \quad \eta = -, \end{cases} \text{ and } \sigma \notin \mathbb{N}. \tag{35}$$

Integral (30) converges since *<sup>λ</sup>*2,*a*(*λ*) tends uniformly to zero as *λ* → ∞ in any fixed half-plane Re *λ δ* > −12 (statement (iii) of Proposition 2); the term exp[−<sup>i</sup>(*σ*+i*μ*)(*<sup>t</sup>*−*ηπ*)] sin *π*(*σ*+i*μ*) is bounded by a constant for *σ* ∈ N (see (35)), and is bounded by 2 as *μ* → ±∞ in view of inequality (34). The contour C can be distorted and replaced by a line *Lσ* parallel to the imaginary axis and crossing the real axis at Re *λ* = *σ* with *σ* −12 (*σ* ∈ N) (see Figure 3) provided the real variable *t* is kept in [0, <sup>2</sup>*π*] for *f*-(+) + (*t*), and in [−2*π*, 0] for *f* -(+) − (*t*), respectively. Note that the integral along *Lσ* converges since *<sup>λ</sup>*,*a*(*λ*) ∈ *<sup>L</sup>*<sup>1</sup>(−∞, <sup>+</sup>∞) for any fixed value of Re *λ* −12 (statement (iv) of Proposition 2) and by inequality (35). We may now apply the Watson resummation to integral (30). For *t* ∈ [0, <sup>2</sup>*π*], we obtain:

$$\frac{1}{4\pi} \int\_{\mathcal{C}} \widetilde{a}(\lambda) \frac{e^{-i\lambda \left(t - \pi\right)}}{\sin \pi \lambda} d\lambda = \frac{1}{2\pi} \sum\_{n=0}^{\infty} a\_n e^{-int} \qquad \left(0 \leqslant t \leqslant 2\pi\right),$$

where the contour C encircles the semi-axis Re *λ* > −12 and crosses it at a point −12 *σ* < 0. Then, distorting the contour C into the line *Lσ* (−12 *σ* < 0), which is admissible as explained above, we obtain for −12 *σ* < 0:

$$\hat{f}\_{+}^{(+)}\left(t\right) = -\frac{1}{4\pi} \int\_{-\infty}^{+\infty} \hat{a}(\sigma + i\mu) \frac{e^{-i(\sigma + i\mu)(t - \pi)}}{\sin \pi(\sigma + i\mu)} d\mu = \frac{1}{2\pi} \sum\_{n=0}^{\infty} a\_{ll} e^{-i n t} \quad \left(0 \le t \le 2\pi\right), \tag{36}$$

and, analogously, for *t* ∈ [−2*π*, 0]:

$$\hat{f}^{(+)}\_{-}(t) = -\frac{1}{4\pi} \int\_{-\infty}^{+\infty} \hat{a}(\sigma + i\mu) \frac{e^{-i(\sigma + i\mu)(t + \pi)}}{\sin \pi(\sigma + i\mu)} d\mu = \frac{1}{2\pi} \sum\_{n=0}^{\infty} a\_n e^{-\text{i}nt} \quad (-2\pi \ll t \ll 0) \,\text{.}\tag{37}$$

**Figure 3.** Integration path of integral (30).

Now, substitute into integral (36) the real variable *t* with the complex variable *τ* = *t* +i*w*. The resulting integral can be proved to provide an analytic continuation of *f* -(+) + (*t*) in the strip 0 < *t* < 2*π*, *w* ∈ R+, continuous in the closure of the latter. In fact, from the first equality in (36), we formally obtain:

$$\hat{f}\_{+}^{(+)}(t+iw) = \frac{1}{2\pi} \, e^{\sigma w} \int\_{-\infty}^{+\infty} H\_{\sigma}^{t}(\mu) \, e^{i\mu w} \, d\mu \qquad \left(0 \lessapprox t \lessapprox 2\pi; w \in \mathbb{R}^{+}, -\frac{1}{2} \lessapprox \sigma < 0\right),\tag{38}$$

where

$$H\_{\sigma}^{t}(\mu) \doteq -\frac{\hat{a}(\sigma + \mathbf{i}\mu)\sigma^{-\mathbf{i}(\sigma + i\mu)(t - \pi)}}{2\sin\pi(\sigma + \mathbf{i}\mu)}.\tag{39}$$

By inequality (35) and statement (vi) of Proposition 2:

$$\left|H\_{\sigma}^{t}(\mu)\right| \leqslant \frac{|\tilde{a}(-\frac{1}{2} + i\mu)|}{|\sin \pi \sigma|} \qquad \left(0 \leqslant t \leqslant 2\pi; \sigma \geqslant -\frac{1}{2}, \sigma \notin \mathbb{N}\right),\tag{40}$$

which, along with statements (iv) and (v) of Proposition 2, guarantees that *<sup>H</sup>tσ*(*μ*) ∈ *<sup>L</sup>*<sup>1</sup>(−∞, <sup>+</sup>∞) for 0 *t* 2*π*, *σ* > −12 , *σ* ∈ N. Therefore, formulas (38), (39), and (40) define *f*-(+) + (*τ*) (*τ* = *t* + i*w*) as an analytic continuation of *f* -(+) + (*t*) in the strip {*τ* = *t* + i*w*, 0 < *t* < 2*π*, *w* ∈ <sup>R</sup><sup>+</sup>}, continuous in the closure of the latter in view of the Riemann–Lebesgue theorem. Proceeding analogously, we can obtain an analytic continuation of *f* -(+) − (*t*) in the strip {*τ* = *t* + i*w*, −2*π* < *t* < 0, *w* ∈ <sup>R</sup><sup>+</sup>}, continuous on the closure of the latter. It then follows that the function *f* -(+)(*τ*) admits a holomorphic extension to the cut-domain *τ*I(0) + \*τ* Ξ˙ (0) + , and statement (ii) is proved.

The discontinuity *F* -(+)(*w*) of (i*f*-(+)(*τ*)) across the cut at *t* = 0 equals i[ *f*-(+) + (<sup>i</sup>*w*) − *f*-(+) − (<sup>i</sup>*w*)] (*w* ∈ R<sup>+</sup>) (for the <sup>2</sup>*π*-periodicity of *f* -(+)(*τ*), we may consider only the cut at *t* = 0). It can be computed by replacing *t* by i*w* into integrals (36) and (37) and then subtracting Equation (37) from Equation (36):

$$\hat{H}^{(+)}(w) \doteq \mathbf{i} \left[ \hat{f}^{(+)}\_{+}(\text{iw}) - \hat{f}^{(+)}\_{-}(\text{iw}) \right] \\ = \frac{1}{2\pi} \int\_{-\infty}^{+\infty} \hat{a}(\sigma + \mathbf{i}\mu) e^{(\sigma + i\mu)w} \, \mathrm{d}\mu \qquad \left(w \in \mathbb{R}^{+}, \sigma \geqslant -\frac{1}{2}\right), \tag{41}$$

which yields:

$$\left|\widehat{F}^{(+)}\left(w\right)\right| \leqslant \left\|\widehat{a}\_{\sigma}\right\|\_{1} \left\|e^{\sigma w}\right\| \qquad \left(w \in \mathbb{R}^{+}, \sigma \geqslant -\frac{1}{2}\right),$$

where ,*aσ*1, defined in (27), is guaranteed to be finite by statement (iv) of Proposition 2. Rewrite (41) as follows:

$$\hat{F}^{(+)}(w)e^{-\sigma w} = \frac{1}{2\pi} \int\_{-\infty}^{+\infty} \hat{a}(\sigma + i\mu) \,\epsilon^{\dagger \mu \nu} \,\mathrm{d}\mu \qquad \left(w \in \mathbb{R}^{+}, \sigma \gtrless -\frac{1}{2}\right). \tag{42}$$

Since for any fixed Re *λ* −12 , *<sup>λ</sup>*,*a*(*λ*) and ,*a*(*λ*) belong to *<sup>L</sup>*<sup>1</sup>(−∞, <sup>+</sup>∞), the Riemann–Lebesgue theorem guarantees that *F* -(+)(*w*)*e*<sup>−</sup>*σ<sup>w</sup>* is a function of class *C*<sup>1</sup> tending to zero as *w* → +<sup>∞</sup>, and statement (iii) is proved. Hence, *F* -(+)(*w*) is a continuous function of *w* (*w* ∈ R<sup>+</sup>) and *<sup>F</sup>*-(+)(*w*) = *o*(*e*<sup>−</sup>*<sup>w</sup>*/2) as *w* → +<sup>∞</sup>. Moreover, *F* -(+)(0) = 0 for the continuity of *f*-(+)(*τ*) on the real axis, and statement (iv) is proved. Inverting (42), we have:

$$\widetilde{a}(\sigma + i\mu) = \int\_0^{+\infty} \widetilde{F}^{(+)}(w) \, e^{-(\sigma + i\mu)w} \, \text{d}w \qquad \left(\sigma > -\frac{1}{2}\right),$$

where the integral on the r.h.s. converges for *σ* > −12 . It defines the Laplace transform of *<sup>F</sup>*-(+)(*w*), holomorphic in the half-plane Re *λ* > −12 , and statement (v) follows. Finally, recalling that ,*a*(*σ* + <sup>i</sup>*μ*) (*μ* ∈ R) belongs to *<sup>L</sup>*<sup>2</sup>(−∞, <sup>+</sup>∞) for any fixed value *σ* −12 (statement (ii) of Proposition 2 and inequality (25)), we obtain the Plancherel equality:

$$\int\_{-\infty}^{+\infty} |\tilde{a}\left(\sigma + i\mu\right)|^2 \,\mathrm{d}\mu = 2\pi \int\_{0}^{+\infty} \left| \hat{F}^{(+)}(w) \, e^{-\sigma w} \right|^2 \,\mathrm{d}w \qquad \left(\sigma \gtrless -\frac{1}{2}\right),$$

proving statement (vi).

**Proposition 4.** *If in the trigonometric series (see* (19)*)*

$$\frac{1}{2\pi} \left[ \sum\_{n=0}^{\infty} a\_{ll} e^{-int} - e^{\text{i}t} \sum\_{n=0}^{\infty} a\_{ll} e^{\text{i}nt} \right] \qquad (t \in \mathbb{R}), \tag{43}$$

*the coefficients an satisfy the assumptions required by Propositions 2 and 3, i.e., the set of numbers fn* = (*n* + <sup>1</sup>)<sup>2</sup>*an satisfies condition* (20)*, then:* -


**Proof.** These statements can be proved by using obvious extensions of the arguments used in the proof of Proposition 3.

Through the holomorphic extension associated with the trigonometric series (43), we obtain a function, denoted by *f* - (*τ*) (*τ* = *t* + i*w*; *t*, *w* ∈ R), which is the analytic continuation of *f* - (*t*) (*t* ∈ R) from the real axis to the domain *τ*I ˙ .= ( *τ*I(0) + \ *τ*Ξ˙ (0) + ) ∪ ( *τ*I(0) − \ *τ*Ξ˙ (0) − ). We can then prove the following corollary to Proposition 4.

**Corollary 1.** *The function f* - (*τ*) *is holomorphic in the* <sup>2</sup>*π-periodic strips* Σ*τ* .= {*τ* ∈ C : *τ* = *t* + i*w*, 2*πk* < *t* < <sup>2</sup>*π*(*k* + <sup>1</sup>), *k* ∈ Z, *w* ∈ <sup>R</sup>}*.*

**Proof.** We start from statement (ii) of Proposition 4. Next, by applying the Schwarz reflection principle and taking into account that the function *f* - (*t*) (i.e., the restriction of *f* - (*τ*) to the real axis) is continuous (statement (i) of Proposition 4), the statement of the corollary follows.

**Remark 2.** *In Ref. [21], the Hausdorff condition* (20) *was assumed to hold for the set of numbers fn* = *npan (p* 2*). However, this condition does not guarantee obtaining a unique Carlsonian interpolation of the whole sequence an, including the first coefficient a*0*. Therefore, it must be replaced by the same condition on the numbers fn* = (*n* + 1)*pan (p* 2*). However, for the purpose of the current analysis, it is sufficient to take p* = 2*. Let us note that the results of Ref. [21] hold true and, in particular, the proofs of Propositions 5.1 and 6.1 of that paper, which will be used below, are correct modulo the following change: fn* = *npan* → *fn* = (*n* + 1)*pan.*

*2.4. Inversion of the Radon–Abel Transformation and Holomorphic Extension Associated with the Legendre Series*

The first step consists of determining the inversion of the Radon–Abel transformation (12). We can prove the following proposition.

**Proposition 5.** *Suppose that the sequence fn* = (*n* +<sup>1</sup>)<sup>2</sup> *an (an being the coefficients of the Legendre expansion* (14)*) satisfies the Hausdorff condition* (20)*. Then, the Radon–Abel transformation (see* (12)*)*

$$\hat{f}(t) = -2\epsilon^{\mathrm{i}t/2} \int\_0^t f(u) \frac{\sin u}{[2(\cos t - \cos u)]^{\frac{1}{2}}} \mathrm{d}u \tag{44}$$

*admits the following inversion:*

$$f(\mu) = \frac{1}{\pi \sin \mu} \frac{\mathrm{d}}{\mathrm{d}\mu} \int\_0^{\mu} e^{-\mathrm{i}t/2} \hat{f}(t) \frac{\sin t}{[2(\cos \mu - \cos t)]^{\frac{1}{2}}} \,\mathrm{d}t.$$

**Proof.** See Proposition 5.1 of Ref. [21].

By following the same procedure used in the proof of Proposition 5 (i.e., introducing Riemann–Liouville integrals and related properties), Proposition 5 can be extended to give the inversion of the Radon–Abel transformation (11). We can state, without proof, the following proposition.

**Proposition 6.** *Assume that the coefficients an (see* (14)*) satisfy the conditions of Proposition 5; then, the function f* - (*τ*)*, i.e., the analytic continuation of the function f* - (*t*) *given in* (44)*, is holomorphic in the strips* Σ*τ and can be represented by the following Radon–Abel transformation (see* (11)*):*

$$\hat{f}(\tau) = -2\,\varepsilon^{\text{i}\tau/2} \int\_{\gamma\_{\tau}} f(\theta) \, \frac{\sin \theta}{[2(\cos \tau - \cos \theta)]^{\frac{1}{2}}} \, \text{d}\theta\text{ }\gamma$$

*which admits the following inversion:*

$$f(\theta) = \frac{1}{\pi \sin \theta} \frac{\mathrm{d}}{\mathrm{d}\theta} \int\_{\gamma\theta} e^{-i\tau/2} \hat{f}(\tau) \frac{\sin \tau}{[2(\cos \theta - \cos \tau)]^{\frac{1}{2}}} \, \mathrm{d}\tau,\tag{45}$$

*γτ and γθ denoting the rays from zero to τ and from zero to θ, respectively.*

We can now prove the following proposition.

**Proposition 7.** *Suppose that the sequence fn* = (*n* + <sup>1</sup>)<sup>2</sup>*an (n* = 0, 1, 2, ...*) satisfies the Hausdorff condition* (20)*, then the function f*(*θ*) *(θ* ∈ C*), represented by formula* (45)*, is even,* <sup>2</sup>*π-periodic, and holomorphic in <sup>θ</sup>*I˙ = ( *<sup>θ</sup>*I(0) +\ *<sup>θ</sup>*Ξ˙ (0) +) ∪ ( *<sup>θ</sup>*I(0) −\ *<sup>θ</sup>*Ξ˙ (0) −)*.*

**Proof.** The assumptions on the Legendre coefficients *an* allow us to state that *f* - (*τ*) is a <sup>2</sup>*π*-periodic function, holomorphic in the domain *τ*I ˙ (see Corollary 1). Moreover, it enjoys the symmetry property:

$$
\hat{f}(\pi) = -e^{i\pi}\hat{f}(-\pi),
$$

which follows from (17) and from the uniqueness of the analytic continuation. The properties mentioned above imply that *f* - (*τ*) is of the form: *f* - (*τ*) = *e*i*τ*/2(<sup>1</sup> − cos *τ*) 12 *b*(cos *<sup>τ</sup>*), with *b*(cos *τ*) analytic in *τD* = {cos *τ* ∈ C, *τ* ∈ *τ*I}˙ . Through the following parametrization of *γθ*: cos *τ* = 1 + *<sup>δ</sup>*(cos *θ* − <sup>1</sup>), (0 *δ* 1), formula (45) can be rewritten as:

$$f(\theta) = \frac{\mathrm{i}}{\sqrt{2}\pi} \frac{\mathrm{d}}{\mathrm{d}(\cos\theta)} \left[ (\cos\theta - 1) \int\_0^1 b(1 + \delta(\cos\theta - 1)) \delta^{\frac{1}{2}} (1 - \delta)^{-\frac{1}{2}} \,\mathrm{d}\delta \right],$$

which represents a function holomorphic in *<sup>θ</sup>D* = {cos *θ* ∈ C, *θ* ∈ *<sup>θ</sup>*I}˙ . Accordingly, regarded as a function of *θ*, it represents an even function, <sup>2</sup>*π*-periodic, and holomorphic in *<sup>θ</sup>*I˙ .

From the previous proposition and Corollary 1, it derives the following corollary.

**Corollary 2.** *If the sequence* {*an*}<sup>∞</sup>*n*=<sup>0</sup> *of the Legendre coefficients satisfies the conditions of Proposition 7, then f*(*θ*) *is a function analytic in the* <sup>2</sup>*π-periodic strips* Σ*θ* .= {*θ* ∈ C : *θ* = *u* + i*v*, 2*πk* < *u* < <sup>2</sup>*π*(*k* + <sup>1</sup>), *k* ∈ Z, *v* ∈ <sup>R</sup>}*.*

Formula (45) allows us to compute the boundary values *f* (+) ± (<sup>i</sup>*v*), which are defined by *f* (+) *η* (*v*) .= lim*u*→0<sup>+</sup> *f* (+)(*η u* + <sup>i</sup>*v*), *η* = ±, *v* 0, on the semi-axis *θ* = i*v*, *v* 0, with *γ*i*v* = {*τ* = i*w*, 0 *w <sup>v</sup>*}, in terms of the corresponding boundary values *f* -(+) ± (<sup>i</sup>*w*), provided *f* -(+) ± (<sup>i</sup>*w*) satisfies a *C*1-type regularity condition; the latter condition is necessary in order to perform the inversion of the Radon–Abel transform at the boundary. The *C*1-continuity of the boundary values follows from the fact that the sequence *fn* = (*n* + <sup>1</sup>)<sup>2</sup>*an* is required to satisfy the Hausdorff condition (20) (see Propositions 2 and 3). We thus obtain:

$$F^{(+)}(\upsilon) \doteq \mathbf{i}[f\_+^{(+)}(\mathrm{i}\upsilon) - f\_-^{(+)}(\mathrm{i}\upsilon)] = \frac{1}{\pi \sinh \upsilon} \frac{\mathrm{d}}{\mathrm{d}\upsilon} \int\_0^\upsilon e^{w/2} \hat{F}^{(+)}(\mathrm{w}) \frac{\sinh w}{[2(\cosh \upsilon - \cosh w)]^{\frac{1}{2}}} \,\mathrm{d}w,\tag{46}$$

where *F* -(+)(*w*) .= i[ *f*-(+) + (<sup>i</sup>*w*) − *f*-(+) − (<sup>i</sup>*w*)], *f*-(+) *η* (<sup>i</sup>*w*) = lim*<sup>t</sup>*→0<sup>+</sup> *f*-(+)(*η t* + <sup>i</sup>*w*), *η* = ±. At this point, let us note that, for the current analysis, it is sufficient to consider the cuts in the *τ*- and *θ*-planes at *τ* = 2*kπ* + i*w* (*k* ∈ Z, *w* > 0) and *θ* = 2*kπ* + i*v* (*k* ∈ Z, *v* > 0), respectively. Therefore, we can limit ourselves to consider the functions: *F* -(+)(*w*) = *e*<sup>−</sup>*<sup>v</sup>*/2(A*F*)(*w*) (*w* 0) and, correspondingly, *<sup>F</sup>*(+)(*v*) ≡ *F*(+)(cosh *v*) (*v* 0). For simplicity, hereafter we shall omit the superscript (+) in these notations.

Next, we can apply the inverse Radon–Abel operator (defined by (45)) to the series on the r.h.s. of formula (19), i.e.,

$$\widehat{f}(t) = \frac{1}{2\pi} e^{i(t-\pi)/2} \sum\_{n=-\infty}^{+\infty} (-1)^n a\_n \cos\left[\left(n + \frac{1}{2}\right)(t-\pi)\right],$$

whose term by term integration is legitimate for the uniform convergence of the series, which follows from the Hausdorff conditions satisfied by the coefficients {*an*}. We now introduce the functions

$$\psi\_n(\cos u) = -\frac{\mathrm{i}}{\pi \sin u} \frac{\mathrm{d}}{\mathrm{d}u} \int\_0^u \cos \left[ \left( n + \frac{1}{2} \right) (t - \pi) \right] \frac{\sin t}{[2(\cos u - \cos t)]^{\frac{1}{2}}} \mathrm{d}t \qquad (0 < u < 2\pi),$$

which are related to the Legendre polynomials *Pn*(cos *u*) by [21]:

$$
\psi\_n(\cos u) = \frac{(-1)^n}{4} (2n+1) \, P\_n(\cos u).
$$

Recalling that *an* = <sup>−</sup>*a*−*n*−1 (*n* ∈ Z) (formula (18)), we finally obtain the original Legendre expansion

$$f(u) = \underline{f}(\cos u) = \frac{1}{\pi} \sum\_{n=0}^{\infty} (-1)^n a\_{\text{ll}} \psi\_{\text{ll}}(\cos u) = \frac{1}{4\pi} \sum\_{n=0}^{\infty} (2n+1) \, a\_{\text{ll}} P\_{\text{ll}}(\cos u).$$

All the results obtained for the function *f*(cos *θ*) in the cos *θ*-plane can be summarized in the following theorem.

**Theorem 2.** *If the sequence fn* = (*n* + <sup>1</sup>)<sup>2</sup>*an (n* = 0, 1, 2, . . .*) satisfies the Hausdorff condition* (20)*, then:*


### *2.5. Spherical Laplace Transform and Analyticity Properties in the Complex λ-Plane*

A basic feature of the holomorphic extensions associated with the series expansions is the *dual analyticity* that we shall illustrate in the specific case that we have considered here, of the Legendre series (14). To classes of functions *f*(cos *θ*) holomorphic in the cos *θ*-plane cut along the semi-axis [1, <sup>+</sup>∞) (see Theorem 2), there correspond classes of analytic functions ,*a*(*λ*) (*λ* ∈ C), which enjoy the following properties:


Formulas (9) and (28) give

$$\widetilde{a}(\lambda) = \int\_0^{+\infty} e^{-\left(\lambda + \frac{1}{2}\right)w} \left(\mathcal{A}F\right)(w) \,\mathrm{d}w \qquad \left(\mathrm{Re}\,\lambda > -\frac{1}{2}\right),$$

which is precisely the spherical Laplace transform, holomorphic in the half-plane Re *λ* > −12 (statement (v) of Proposition 3). Moreover, ,*<sup>a</sup>*(*λ*)|*<sup>λ</sup>*=*<sup>n</sup>* = *an* (statement (i) of Proposition 2). Writing explicitly the Abel transform (A*F*)(*w*) (see (9)), we obtain

$$\widetilde{a}(\lambda) = 2 \int\_0^{+\infty} e^{-(\lambda + \frac{1}{2})w} \left\{ \int\_0^w \underline{E}(\cosh v) \frac{\sinh v}{[2(\cosh w - \cosh v)]^{\frac{1}{2}}} \, \mathrm{d}v \right\} \, \mathrm{d}w \quad \left(\mathrm{Re}\, \lambda > -\frac{1}{2}\right),$$

which, interchanging the order of integration, becomes

$$\widetilde{u}(\lambda) = 2 \int\_0^{+\infty} \underline{F}(\cosh v) \sinh v \left\{ \int\_v^{+\infty} \frac{e^{-(\lambda + \frac{1}{2})w}}{[2(\cosh w - \cosh v)]^{\frac{1}{2}}} \, \text{d}w \right\} \, \text{d}v \quad \left(\text{Re}\, \lambda > -\frac{1}{2}\right). \tag{47}$$

Using the integral representation of the Legendre functions of the second kind [24]

$$Q\_{\lambda}(\cosh v) = \int\_{\upsilon}^{+\infty} \frac{e^{-(\lambda + \frac{1}{2})w}}{[2(\cosh w - \cosh v)]^{\frac{1}{2}}} \, \text{d}w \qquad (\text{Re}\,\lambda > -1, \upsilon > 0) \, \tag{48}$$

formula (47) can be written as follows:

$$\widetilde{u}(\lambda) = 2 \int\_0^{+\infty} \underline{F}(\cosh v) \, Q\_{\lambda}(\cosh v) \sinh v \, \mathrm{d}v \qquad \left(\mathrm{Re}\,\lambda > -\frac{1}{2}\right). \tag{49}$$

**Remark 3.** *The Legendre function of the second kind has a logarithmic singularity at v* = 0*, then the integral representation* (48) *holds if v* > 0*; nevertheless, the integral in* (49) *converges if F*(cosh *v*) *is regular at v* = 0*.*

We can now state the following theorem.

**Theorem 3.** *If the sequence fn* = (*n* + <sup>1</sup>)<sup>2</sup>*an (n* = 0, 1, 2, ...*), an being the Legendre coefficients of expansion* (14)*, satisfies the Hausdorff condition* (20)*, then the jump function <sup>F</sup>*(*v*) = *F*(cosh *v*) *(defined in* (46)*) admits the following integral representation:*

$$\begin{split} F(\boldsymbol{\nu}) &= \mathbb{E}(\cosh \boldsymbol{\nu}) = \frac{1}{2\pi} \int\_{-\infty}^{+\infty} \widetilde{\boldsymbol{a}}(\sigma + \mathbf{i}\mu) \left(\sigma + \frac{1}{2} + \mathbf{i}\mu\right) P\_{\sigma + i\mu}(\cosh \boldsymbol{\nu}) \, \mathrm{d}\mu \\ &= \frac{1}{2\pi \mathrm{i}} \int\_{\sigma - i\infty}^{\sigma + i\infty} \widetilde{\boldsymbol{a}}(\lambda) \left(\lambda + \frac{1}{2}\right) P\_{\lambda}(\cosh \boldsymbol{\nu}) \, \mathrm{d}\lambda \qquad \left(\lambda = \sigma + \mathrm{i}\mu; \sigma \geqslant -\frac{1}{2}\right), \end{split} \tag{50}$$

*where <sup>P</sup>λ*(·) *denotes the Legendre function of the first kind.*

### **Proof.** See Proposition 6.1 of Ref. [21].

### **3. From Spherical Laplace Transform to Non-Euclidean Fourier Transform**

### *3.1. Formal Derivation of Mehler's Transform from the Spherical Laplace Transform*

Let us begin by considering the integral representation (48) of the Legendre function of the second kind *Qλ*(cosh *<sup>v</sup>*). If Re *λ* = −12 , the function *Q*− 12 <sup>+</sup>i*μ*(cosh *v*) can be written as the sum of its even (in *μ*) and odd parts, which are defined as follows:

$$\mathcal{Q}^{(\rm E)}\_{-\frac{1}{2}+i\mu}(\cosh v) = \int\_{\upsilon}^{+\infty} \frac{\cos \mu w}{[2(\cosh w - \cosh \upsilon)]^{\frac{1}{2}}} \, \mathrm{d}w$$

$$\mathcal{Q}^{(\rm O)}\_{-\frac{1}{2}+i\mu}(\cosh \upsilon) = -i \int\_{\upsilon}^{+\infty} \frac{\sin \mu w}{[2(\cosh w - \cosh \upsilon)]^{\frac{1}{2}}} \, \mathrm{d}w$$

Recalling the relation between the Legendre functions of first and second kind (p. 140, [24])

$$P\_{\lambda}(z) = \frac{\tan \pi \lambda}{\pi} \left[ Q\_{\lambda}(z) - Q\_{-\lambda - 1}(z) \right],$$

and exploiting the evenness (in *μ*) of the conical Legendre functions, i.e., *P*− 12 <sup>+</sup>i*μ*(cosh *v*) = *P*− 12 −i*μ*(cosh *v*) (*μ* ∈ R), we obtain:

$$P\_{-\frac{1}{2}+i\mu}(\cosh v) = P\_{-\frac{1}{2}-i\mu}(\cosh v) = \frac{2}{\pi}\tan\left[\pi\left(-\frac{1}{2}+i\mu\right)\right]Q\_{-\frac{1}{2}+i\mu}^{(0)}(\cosh v). \tag{51}$$

Let us now return to Theorem 3 and to formula (50) in the specific case *σ* = −12 . By the *μ*-evenness of *P* − 1 2 <sup>+</sup>i*μ*(cosh *<sup>v</sup>*), it follows that only the odd component ,*a* (O)(−12 + <sup>i</sup>*μ*) of ,*<sup>a</sup>*(−12 + <sup>i</sup>*μ*) (*μ* ∈ R) contributes to the integral in formula (50):

$$F(\upsilon) = \underline{F}(\cosh \upsilon) = \frac{\mathsf{i}}{\pi} \int\_0^{+\infty} \check{a}^{(\mathsf{O})} \left( -\frac{1}{2} + \mathsf{i}\mu \right) P\_{-\frac{1}{2} + i\mu}(\cosh \upsilon) \,\mu \,\mathrm{d}\mu. \tag{52}$$

Moving back to formula (49), which for the moment we assume to hold also for Re *λ* = −12 (this statement will be proved later in Section 3.4.3), and accounting for relationship (51), we can *formally* rewrite the odd component of ,*<sup>a</sup>*(−12 + <sup>i</sup>*μ*) (*μ* ∈ R) as follows:

$$\hat{a}^{(\mathcal{O})}\left(-\frac{1}{2} + i\mu\right) = \frac{\pi}{\tan\left[\pi\left(-\frac{1}{2} + i\mu\right)\right]} \int\_0^{+\infty} \underline{F}(\cosh v) \, P\_{-\frac{1}{2} + i\mu}(\cosh v) \sinh v \, \mathrm{d}v \qquad (\mu \in \mathbb{R}).\tag{53}$$

Noting that (tan[*π*(−12 + <sup>i</sup>*μ*)])−<sup>1</sup> = −i tanh *πμ*, we may introduce the following function: *<sup>F</sup>*,(*μ*) .= ,*<sup>a</sup>*(−12 + <sup>i</sup>*μ*)/(−i*<sup>π</sup>* tanh *πμ*) (we omit here the superscript "(O)" in ,*a*(O) −12 + i*μ* since only the odd part of , *<sup>a</sup>*(−12 + <sup>i</sup>*μ*) plays a role in these transformations). Then, formulas (52) and (53) can be rewritten as

$$F(v) = \underline{F}(\cosh v) = \int\_0^{+\infty} \hat{F}(\mu) \, P\_{-\frac{1}{2} + i\mu}(\cosh v) \, \tanh(\pi \mu) \, \mu \, \mathrm{d}\mu,\tag{54a}$$

$$\widetilde{F}(\mu) = \int\_0^{+\infty} \mathbb{E}(\cosh v) \, P\_{-\frac{1}{2} + i\mu}(\cosh v) \sinh v \, \mathrm{d}v,\tag{54b}$$

which coincide with the Mehler transform pair, indeed (see (p. 175, [24])).

**Remark 4.** *(i) As we have already remarked above, we do not consider the whole spherical Laplace transform but only its odd (with respect to μ) component since only this component plays a role in the transformations being treated here.*

*(ii) The class of functions <sup>F</sup>*(*v*) *which we are led to consider here are of the form F* = *F*(cosh *<sup>v</sup>*)*. Accordingly, the connection between spherical Laplace transform and non-Euclidean Fourier transform can be limited to this class of functions. In the non-Euclidean geometry, these functions belong to the class of* radial functions *in a sense that will be clarified in what follows (see Remarks 5 and 6).*

### *3.2. Geometry of the Two-Sheeted Hyperboloid: Polar and Horocyclic Coordinates*

Let us now return to the geometrical representation of the real one-sheeted hyperboloid *X*2. It can be easily noted that, by simply swapping two coordinate axes (see Figures 1 and 4), the real meridian *X* -+ can be regarded as the real meridian of one of the sheets of a suitable real two-sheeted hyperboloid. It is therefore reasonable to expect that an integral representation of *F*(cosh *v*) can be obtained also from the geometry of the two-sheeted hyperboloid.

In the space R<sup>3</sup> of variables *x* = (*<sup>x</sup>*0, *x*1, *<sup>x</sup>*2), we consider the two-sheeted hyperboloid with equation:

$$
\lambda x\_0^2 - x\_1^2 - x\_2^2 = 1.\tag{55}
$$

In the present analysis, we can limit ourselves to consider the upper sheet of this two-sheeted hyperboloid, the one with *x*0 1 that will be denoted by <sup>2</sup>*X*+2 . By analogy with what we have done is Section 2, we consider two systems of coordinates: polar and horocyclic coordinates.
