Harmonic Analysis on the Möbius Gyrogroup

Abstract

In this paper we propose to develop harmonic analysis on the Poincaré ball ${{\mathbb {B}}_{t}^{n}}$, a model of the $n$-dimensional real hyperbolic space. The Poincaré ball ${{\mathbb {B}}_{t}^{n}}$ is the open ball of the Euclidean $n$-space $\mathbb {R}^n$ with radius $t >0$, centered at the origin of $\mathbb {R}^n$ and equipped with Möbius addition, thus forming a Möbius gyrogroup where Möbius addition in the ball plays the role of vector addition in $\mathbb {R}^n.$ For any $t>0$ and an arbitrary parameter $\sigma \in \mathbb {R}$ we study the $(\sigma ,t)$-translation, the $(\sigma ,t)$-convolution, the eigenfunctions of the $(\sigma ,t)$-Laplace–Beltrami operator, the $(\sigma ,t)$-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 $t \rightarrow +\infty$ the resulting hyperbolic harmonic analysis on ${{\mathbb {B}}_{t}^{n}}$ tends to the standard Euclidean harmonic analysis on $\mathbb {R}^n,$ thus unifying hyperbolic and Euclidean harmonic analysis. As an application we construct diffusive wavelets on ${{\mathbb {B}}_{t}^{n}}$.

Appendices

Appendix 1: Spherical Harmonics

A spherical harmonic of degree $k\ge 0$ denoted by $Y_k$ is the restriction to ${\mathbb {S}}^{n-1}$ of a homogeneous harmonic polynomial in $\mathbb {R}^n.$ The set of all spherical harmonics of degree $k$ is denoted by $\mathcal {H}_k({\mathbb {S}}^{n-1}).$ This space is a finite dimensional subspace of $L^2({\mathbb {S}}^{n-1})$ and we have the direct sum decomposition

$$\begin{aligned} L^2({\mathbb {S}}^{n-1}) = \bigoplus _{k=0}^\infty \mathcal {H}_k({\mathbb {S}}^{n-1}). \end{aligned}$$

The following integrals are obtained from the generalisation of Lemma 2.4 in [17].

Lemma 5

[17] Let $\nu \in \mathbb {C}, k \in \mathbb {N}_0, t \in \mathbb {R}^+,$ and $Y_k \in \mathcal {H}_k({\mathbb {S}}^{n-1}).$ Then

$$\begin{aligned} \displaystyle \int _{{\mathbb {S}}^{n-1}} \frac{Y_k(\xi )}{\left\| \frac{x}{t}-\xi \right\| ^{2\nu }}~d\sigma (\xi ) \!=\! \frac{(\nu )_k}{(n/2)_k}\,{}_2F_1 \left( k\!+\!\nu , \nu \!-\!\frac{n}{2}\!+\!1; k\!+\!\frac{n}{2}; \frac{\Vert x\Vert ^2}{t^2}\right) \frac{\Vert x\Vert ^k}{t^k}Y_k(x') \end{aligned}$$

where $x \in {{\mathbb {B}}_{t}^{n}}, x'=\Vert x\Vert ^{-1}x,$ $(\nu )_k$ denotes the Pochammer symbol, and $d\sigma$ is the normalised surface measure on ${\mathbb {S}}^{n-1}.$ In particular, when $k=0,$ we have

$$\begin{aligned} \displaystyle \int _{{\mathbb {S}}^{n-1}} \frac{1}{\left\| \frac{x}{t}-\xi \right\| ^{2\nu }}~d\sigma (\xi ) ={}_2F_1 \left( \nu , \nu -\frac{n}{2}+1; \frac{n}{2}; \frac{\Vert x\Vert ^2}{t^2}\right) . \end{aligned}$$

(81)

The Gauss Hypergeometric function ${}_2F_1$ is an analytic function for $|z|<1$ defined by

$$\begin{aligned} \displaystyle {}_2F_1(a,b;c;z)= \displaystyle \sum _{k=0}^{\infty } \frac{(a)_k(b)_k}{(c)_k}\frac{z^k}{k!} \end{aligned}$$

with $c \notin -\mathbb {N}_0.$ If $\text {Re}(c-a-b)>0$ and $c \notin -{\mathbb N}_0$ then exists the limit $\displaystyle \lim _{t\rightarrow 1^{-}} {}_2F_1(a,b;c;t)$ and equals

$$\begin{aligned} {}_2F_1(a,b;c;1) = \displaystyle \frac{\Gamma (c)\Gamma (c-a-b)}{\Gamma (c-a)\Gamma (c-b)}. \end{aligned}$$

(82)

Some useful properties of this function are

$$\begin{aligned} {}_2F_1(a,b;c;z) = (1-z)^{c-a-b}\,{}_2F_1(c-a,c-b;c;z) \end{aligned}$$

(83)

$$\begin{aligned} {}_2F_1(a,b;c;z) = (1-z)^{-a}\,{}_2F_1\left( a,c-b;c;\frac{z}{z-1}\right) \end{aligned}$$

(84)

$$\begin{aligned} \frac{d}{dz}\,{}_2F_1(a,b;c;z) = \frac{ab}{c}\,{}_2F_1(a+1,b+1;c+1;z). \end{aligned}$$

(85)

Appendix 2: Jacobi Functions

The classical theory of Jacobi functions involves the parameters $\alpha ,\beta ,\lambda \in \mathbb {C}$ (see [14, 15]). Here we introduce the additional parameter $t \in \mathbb {R}^+$ since we develop our hyperbolic harmonic analysis on a ball of arbitrary radius $t.$ For $\alpha ,\beta ,\lambda \in \mathbb {C},$ $t \in \mathbb {R}^+,$ and $\alpha \ne -1,-2,\ldots ,$ we define the Jacobi transform as

$$\begin{aligned} \mathcal {J}_{\alpha ,\beta }g(\lambda t) = \int _0^{+\infty } g(r) \varphi _{\lambda t}^{(\alpha ,\beta )}(r) \,\omega _{\alpha ,\beta }(r)~dr \end{aligned}$$

(86)

for all functions $g$ defined on $\mathbb {R}^+$ for which the integral (86) is well defined. The weight function $\omega _{\alpha ,\beta }$ is given by

$$\begin{aligned} \omega _{\alpha ,\beta }(r) = (2\sinh (r))^{2\alpha +1}(2\cosh (r))^{2\beta +1} \end{aligned}$$

and the function $\varphi _{\lambda t}^{(\alpha ,\beta )}(r)$ denotes the Jacobi function which is defined as the even $C^\infty$ function on $\mathbb {R}$ that equals $1$ at $0$ and satisfies the Jacobi differential equation

$$\begin{aligned}&\left( \frac{d^2}{dr^2} + ((2\alpha +1)\coth (r) + (2\beta +1)\tanh (r))\frac{d}{dr} + (\lambda t)^2 + (\alpha +\beta +1)^2 \right) \\&\quad \varphi _{\lambda t}^{(\alpha ,\beta )}(r)=0. \end{aligned}$$

The function $\varphi _{\lambda t}^{(\alpha ,\beta )}(r)$ can be expressed as an hypergeometric function

$$\begin{aligned} \varphi _{\lambda t}^{(\alpha ,\beta )}(r) = {}_2F_1 \left( \frac{\alpha + \beta +1 + {\mathrm {i}}\lambda t}{2}, \frac{\alpha + \beta +1 - {\mathrm {i}}\lambda t}{2}; \alpha +1; - \sinh ^2(r)\right) . \end{aligned}$$

(87)

Since $\varphi _{\lambda t }^{(\alpha ,\beta )}$ are even functions of $\lambda t \in \mathbb {C}$ then $\mathcal {J}_{\alpha ,\beta }g(\lambda t)$ is an even function of $\lambda t$. Payley-Wiener Theorem and some inversion formulas for the Jacobi transform are found in [15]. We denote by $C^{\infty }_{0,R}(\mathbb {R})$ the space of even $C^\infty$-functions with compact support on $\mathbb {R}$ and $\mathcal {E}$ the space of even and entire functions $g$ for which there are positive constants $A_g$ and $C_{g,n}, n=0,1,2,\ldots ,$ such that for all $\lambda \in \mathbb {C}$ and all $n=0,1,2,\ldots$

$$\begin{aligned} |g(x)| \le C_{g,n}(1+|\lambda |)^{-n}\,e^{A_g|\text {Im}(\lambda )|} \end{aligned}$$

where $\text {Im}(\lambda )$ denotes the imaginary part of $\lambda .$

Theorem 9

([15, p. 8]) (Payley–Wiener Theorem) For all $\alpha ,\beta \in \mathbb {C}$ with $\alpha \ne -1,-2,\ldots$ the Jacobi transform is bijective from $C^{\infty }_{0,R}(\mathbb {R})$ onto $\mathcal {E}.$

The Jacobi transform can be inverted under some conditions [15]. Here we only refer to the case which is used in this paper.

Theorem 10

([15, p. 9]) Let $\alpha ,\beta \in \mathbb {R}$ such that $\alpha >-1, \alpha \pm \beta +1 \ge 0.$ Then for every $g \in C^{\infty }_{0,R}(\mathbb {R})$ we have

$$\begin{aligned} g(r) = \frac{1}{2\pi }\int _0^{+\infty } (\mathcal {J}_{\alpha ,\beta }g)(\lambda t) ~ \varphi _{\lambda t}^{(\alpha ,\beta )}(r)~|c_{\alpha ,\beta }(\lambda t)|^{-2} ~t ~ d\lambda , \end{aligned}$$

(88)

where $c_{\alpha ,\beta }(\lambda t)$ is the Harish-Chandra $c$-function associated to $\mathcal {J}_{\alpha ,\beta }(\lambda t)$ given by

$$\begin{aligned} c_{\alpha ,\beta }(\lambda t)= \frac{2^{\alpha +\beta +1-{\mathrm {i}}\lambda t} \Gamma (\alpha +1)\Gamma ({\mathrm {i}}\lambda t)}{\Gamma \left( \frac{\alpha +\beta +1+{\mathrm {i}}\lambda t}{2} \right) \Gamma \left( \frac{\alpha -\beta +1+{\mathrm {i}}\lambda t}{2} \right) }. \end{aligned}$$

(89)

This theorem provides a generalisation of Theorem 2.3 in [15] for arbitrary $t \in \mathbb {R}^+.$ From [15] and considering $t \in \mathbb {R}^+$ arbitrary we have the following asymptotic behavior of $\phi _{\lambda t}^{\alpha ,\beta }$ for $\text {Im}(\lambda )<0:$

$$\begin{aligned} \lim _{r \rightarrow +\infty } \varphi _{\lambda t}^{(\alpha ,\beta )}(r) e^{(-{\mathrm {i}}\lambda t +\alpha +\beta +1)r}= c_{\alpha ,\beta }(\lambda t). \end{aligned}$$

(90)