the journal of fourier analysis and applications · the journal of fourier analysis and...

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706.

1956

v1 [

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13

Jun

2007

The Journal of Fourier Analysis and Applications

Spherical Continuous WaveletTransforms arising from sections

of the Lorentz group

Milton Ferreira

ABSTRACT. In this paper we consider the conformal group of the unit

sphere Sn−1, the proper Lorentz group Spin+(1, n), for the study of spherical

continuous wavelet transforms. The parameter space is determined by the fac-

torization of the gyrogroup of the unit ball by an appropriate gyro-subgroup. We

study two families of sections that give rise to anisotropic conformal dilation

operators for the unit sphere Sn−1 in Rn

associated to Mobius transforma-

tions. Afterwards we show that we can construct wavelets on the unit sphere

from wavelets on the tangent plane.

1. Introduction

The construction of integral transforms related to group representations onL2−spaces on manifolds is a non-trivial problem because the square integra-bility property may fail to hold. For example, in the case of the unit sphereSn−1 there are no square integrable unitary representations on L2(S

n−1) forn > 3 (see [14]). A way to overcome this fact is to make the group smaller,i.e., to factor out a suitable closed subgroup H. In this way we restrict therepresentation to a quotient X = G/H. This approach was successful real-ized in the case of the unit sphere ([3], [4]). The authors used the conformalgroup of Sn−1, the connected component of the Lorentz group SOo(1, n) andits Iwasawa decomposition, or KAN−decomposition, where K is the max-imal compact subgroup SO(n), A = SO(1, 1) ∼= (R+

∗ ,×) is the subgroupof Lorentz boosts in the xn−direction and N ∼= R

n−1 is a nilpotent sub-group. The factorization of SOo(1, n) by N yields the homogeneous space

Math Subject Classifications. Primary: 42C40; Secondary 20N05, 30G35.Keywords and Phrases. spherical continuous wavelet transform, gyrogroups, homo-geneous spaces, sections

c© 2004 Birkhauser Boston. All rights reservedISSN 1069-5869 DOI: jfaatest- 05/28/04

http://arxiv.org/abs/0706.1956v1

2 Milton Ferreira

X = SOo(1, n)/N ∼= SO(n) × R+∗ that gives rise to the parameter space of

the spherical continuous wavelet transform (SCWT) developed in [4].

In physics, the Lorentz group is the classical setting for all (nongravit-tional) physical phenomena. The mathematical form of the kinematical lawsof special relativity, Maxwell’s field equations in the theory of electromag-netics and Dirac’s equation in the theory of the electron are each invariantunder Lorentz transformations. In pure mathematics, the restricted Lorentzgroup arises as the Mobius group, which is the symmetry group of conformalgeometry on the Riemann sphere.

Taking into account these observations we would like to extend theSCWT of Antoine and Vandergheynst. One of its limitations is that itdoes not incorporate relativistic movements on the sphere. But, in manyapplications such movements are required, e.g., an observer who is moving atrelativistically velocity with respect to the Earth would see the appearanceof the night sky (as modeled by points on the celestial sphere) transformedby a Mobius transformation. From the mathematical point of view ourapproach allow us to connect the geometry of conformal transformations onthe sphere with wavelet theory. Another motivation comes from the caseof the plane where now a collection of different types of wavelets, such ascurvelets or shearlets, exists (see e.g. [13]). For a future consideration ofsuch transformations on the sphere it seems to be necessary to incorporategeneral conformal transformations first. From the physical point of view wewill obtain relativistic coherent states which can be applied to the study andrepresentation of massive fields, as well as the study of the cosmic microwavebackground.

The set Dv of all relativistically admissible velocities is a ball in R3

of radius c, the speed of light. It is a homogeneous ball and a boundedsymmetric domain with respect to the group of projective transformations.Relativistic dynamics is described by elements of the Lie Algebra of thisgroup ([10]). It is this normalized homogeneous ball that is on the basis ofour work since it encodes all the information needed for our purpose.

The group Spin+(1, n) (double covering group of SO0(1, n)) togetherwith its Cartan decomposition constitutes a very rich and powerful modelfor the description of the spherical continuous wavelet transform with a nicegeometric description. This rests on the study of the properties of sphericalcaps under the action of Mobius transformations (see [5]), which is in thespirit of the Erlangen program of F. Klein, the study of the invariants of agiven geometry.

The paper is organized as follows. In section 2 we present the notions ofClifford algebras needed to the readability of the paper. More details on thistopic can be found e.g. in [9]. In section 3 we describe the proper Lorentzgroup Spin+(1, n) in the framework of Clifford analysis and we introduce thestructure of gyrogroup necessary to the description of (Bn,⊕). In section 4

Spherical Continuous Wavelet Transforms arising from sections of the Lorentz group 3

we factorize the gyrogroup (Bn,⊕) by the gyro-subgroup (Dn−1en ,⊕) and

we construct sections for the respective homogeneous space, which can beextended to the whole of the group. In section 5 we present the wavelettheory associated to homogeneous spaces and in section 6 we apply thetheory to our case. Finally, in section 7 for the sake of simplicity we restrictourselves to the sphere S2 and we relate wavelets on S2 with wavelets on itstangent plane.

2. Preliminaries

Let e1, . . . , en be an orthonormal basis of Rn. The universal real Clifford

algebra Cl0,n is the free algebra over Rn generated modulo the relation

x2 = −|x|2, for x ∈ Rn. The non-commutative multiplication in Cl0,n is

governed by the rules

eiej + ejei = −2δi,j, ∀ i, j ∈ {1, . . . , n}.

For a set A = {i1, . . . , ih} ⊂ {1, . . . , n}, with 1 ≤ i1 < . . . < ih ≤ n, leteA = ei1ei2 · · · eih and e∅ = 1. Then {eA : A ⊂ {1, . . . , n}} is a basis forCl0,n. Thus, any a ∈ Cl0,n may be written as a =

∑A aAeA, with aA ∈ R

or still as a =∑n

k=0[a]k, where [a]k =∑

|A|=k aAeA is a so-called k-vector(k = 0, 1, . . . , n).

We can identify a vector x = (x1, . . . , xn) ∈ Rn with the one-vector

x =∑n

j=1 xjej. The product of any two vectors (or geometric product) isgiven by a scalar part and a bivectorial part

xy = −〈x, y〉+ x ∧ y

where

〈x, y〉 =m∑

j=1

xjyj = −1

2(xy + yx)

and

x ∧ y =∑

i<j

(xiyj − xjyi)eij =1

2(xy − yx).

Conjugation in Cl0,n is defined as an anti-involution for which ej = −ej ,j = 1, . . . , n, 1 = 1, and ab = b a for any a, b ∈ Cl0,n. In particular for avector x we have x = −x. Each non-zero vector x ∈ R

n is invertible withx−1 := x

|x|2 . Due to the non-commutative character of Clifford algebras, the

product by the inverse at left is in general different from the product by theinverse at right. We denote by x

y the product xy−1, there is, by means ofthe right-hand side inverse.

The Spin group Spin(n) = {∏2mi=1 wi, wi ∈ Sn−1,m ∈ N} arises as a

multiplicative group. For each s ∈ Spin(n), the mapping χ(s) : x 7→ χ(s)x =

4 Milton Ferreira

sxs is a special orthogonal transformation, a rotation. As kerχ = {−1, 1}the group Spin(n) is a double covering group of SO(n).

From now on we identify e1, . . . , en with the canonical basis in Rn.

3. The proper Lorentz group Spin+(1, n)

The full Lorentz group G = SO(1, n) consists of linear homogeneous trans-formations of the (n + 1)−dimensional space under which the quadraticform |x|2 = x20 − |x|2, x = (x0, x), x = (x1, . . . , xn) is invariant. Here weidentify x0 as the time component and the n−vector x = (x1, x2, . . . , xn)as the spatial component. The group of all Lorentz transformations pre-serving both orientation and the direction of time is called the proper or-thochronous Lorentz group and it is denoted by SO0(1, n) or sometimes bySO+(1, n). It is generated by spatial rotations of the maximal compact sub-group K = SO(n) and hyperbolic rotations of the subgroup A = SO(1, 1),accordingly to the Cartan decomposition KAK of SO0(1, n) (see [14], [18]).

The group SO0(1, n) is connected and locally compact. The coset spaceX = SO0(1, n)/K is the Lobachevsky space of n dimensions. It can berealized in various manners, e.g. by the upper sheet of the hyperboloidH+ = {x = (x0, x) : x

20 − |x|2 = 1, x0 > 0}. However, for our purpose, we

shall use throughout this paper the realization using the interior of the unitsphere Sn−1, i.e. Bn = {x ∈ R

n : |x| < 1}.The double covering group of SO0(1, n) is the group Spin+(1, n). In

Clifford Analysis it can be described by Vahlen matrices (see [6], [12]). Thesematrices can be decomposed into the maximal compact subgroup Spin(n)and the set of Mobius transformations of the form ϕa(x) = (x − a)(1 +ax)−1, a ∈ Bn, which map the closed unit ball Bn onto itself. These arethe multi-dimensional analogous transformations of the well known Mobiustransformations of the complex plane ϕu(z) =

z−u1+uz , z ∈ C, u ∈ D, where D

denotes the open unit disc in C.Mobius transformations ϕa(x) can also be written as

ϕa(x) =(1− |a|2)x− (1 + |x|2 − 2 〈a, x〉)a

1− 2 〈a, x〉+ |a|2|x|2 . (3.1)

The composition of two Mobius transformations of type (3.1) is (up toa rotation) again a Mobius transformation of the same type:

ϕa ◦ ϕb = q ϕ(1−ab)−1(a+b)(x) q, q =1− ab

|1− ab| ∈ Spin(n). (3.2)

Thus, we can endow the unit ball Bn with a binary operation

b⊕ a = (1− ab)−1(a+ b) = (a+ b)(1 − ba)−1 = ϕ−b(a). (3.3)


We notice that on (Bn,⊕) the neutral element is zero while each elementa ∈ Bn has an inverse given by the symmetric element −a. This operationis non-commutative since

b⊕ a = q (a⊕ b) q, q =1− ab

|1− ab| ∈ Spin(n), (3.4)

and it is also non-associative due to the presence again of the rotation in-duced by q.

Proposition 1. For all a, b, c ∈ Bn it holds

a⊕ (b⊕ c) = (a⊕ b)⊕ (qcq), with q =1− ab

|1− ab| ∈ Spin(n). (3.5)

Proof. By the associativity of the composition of Mobius transformationswe have on the one hand

((ϕc ◦ ϕb) ◦ ϕa)(x) = ((q1ϕb⊕cq1) ◦ ϕa)(x)= q1ϕb⊕c(ϕa(x))q1= q1q2ϕa⊕(b⊕c)(x)q2 q1

= q1q2ϕa⊕(b⊕c)(x)q2 q1,

with q1 =1−cb|1−cb| and q2 =

1−(b⊕c)a|1−(b⊕c)a| .

On the other hand we have

(ϕc ◦ (ϕb ◦ ϕa))(x) = (ϕc ◦ (q3ϕa⊕bq3))(x)= ϕc(q3ϕa⊕b(x)q3)

= q3ϕq3cq3(ϕa⊕b(x))q3= q3q4ϕ(a⊕b)⊕(q3cq3)(x)q4 q3,

with q3 =1−ba|1−ba| and q4 =

1−(q3cq3)(a⊕b)|1−(q3cq3)(a⊕b)| .

Thus, q1q2 = q3q4 and a⊕ (b⊕ c) = (a⊕ b)⊕ (qcq), with q = 1−ab|1−ab| .

The associativity of ⊕ occurs only in some special cases as we can seein the next Lemma.

Lemma 1. If a, b, c ∈ Bn such that a//b or a ⊥ c and b ⊥ c then theoperation ⊕ is associative, i.e.

a⊕ (b⊕ c) = (a⊕ b)⊕ c.

Left and right cancelation laws have different properties:

6 Milton Ferreira

Lemma 2. For all a, b ∈ Bn it holds

(−b)⊕ (b⊕ a) = a (3.6)

(a⊕ b)⊕ (q(−b)q) = a, with q =1− ab

|1− ab| . (3.7)

Thus, (Bn,⊕) does not have a classic group structure but a structureof a gyrogroup. This structure was introduced by A. Ungar ([15], [16], [17]).Gyrogroups are special loops which share remarkable analogies with groups,thus generalizing the notion of group. The first known gyrogroup is therelativistic gyrogroup (B3,⊕), that appeared in 1988 [15]. It consists of theunit ball B3 = {x ∈ R

3 : |x| < 1} of the Euclidean 3−space, with Ein-stein’s addition ⊕ of relativistically admissible velocities. The operation ⊕is neither commutative nor associative. The notion of gyrogroup appears bythe extension of the Einstein relativistic groupoid (B3,⊕) and the Thomasprecession effect (related by the rotation induced by q). The gyrolanguagewas adopted by Ungar since 1991. It rests on the unification of Euclideanand hyperbolic geometry in terms of the analogies shared [17].

Definition 1. (see [17]) A groupoid (G,⊕) is a gyrogroup if its binaryoperation satisfies the following axioms:

(G1) In G there is at least one element 0, called a left identity satisfying0⊕ a = a, for all a ∈ G.

(G2) For each a ∈ G there is an element ⊖a ∈ G, called a left inverse ofa, satisfying ⊖a⊕ a = 0.

(G3) For any a, b, c, d ∈ G there exists a unique element gyr[a, b]c ∈ Gsuch that the binary operation satisfies the left gyroassociative law

a⊕ (b⊕ c) = (a⊕ b)⊕ gyr[a, b]c.

(G4) The map gyr[a, b] : G → G given by c 7→ gyr[a, b]c is an automor-phism of the groupoid (G,⊕), that is gyr[a, b] ∈ Aut(G,⊕).

(G5) The gyroautomorphism gyr[a, b] possesses the left loop property

gyr[a, b] = gyr[a⊕ b, b].

Definition 2. A gyrogroup (G,⊕) is gyrocommutative if its binary oper-ation satisfies a⊕ b = gyr[a, b](b⊕ a), for all a, b ∈ G.

One defines left gyrotranslations as b 7→ a⊕b and right gyrotranslationsas b 7→ b⊕ a. However, they have different properties.

Proposition 2. ([17]) Let (G,⊕) be a gyrogroup and let a, b ∈ G. Theunique solution of the equation a⊕x = b in G is given by x = ⊖a⊕b, and theunique solution of the equation x⊕a = b in G is given by x = b⊖ gyr[b, a]a.


One of the most important results of this theory is that the gyro-semidirect product of a gyrogroup (G,⊕) with a gyroautomorphism groupH ⊂ Aut(G,⊕) is a group [17]. In our case the result can be stated as

Proposition 3. The gyro-semidirect product between (Bn,⊕) and Spin(n)is the group of pairs (s, a) where a ∈ Bn and s ∈ Spin(n), with operation ×given by the gyro-semidirect product

(s1, a)× (s2, b) = (s1s2q, b⊕ (s2as2)), with q =1− s2as2b

|1− s2as2b|. (3.8)

This is a generalization of the external semidirect product of groups (see[11]). We need to establish some properties between Mobius transformationsand rotations.

Lemma 3. For s ∈ Spin(n) and a ∈ Bn we have

(i) ϕa(sxs) = sϕsas(x)s and (ii) sϕa(x)s = ϕsas(sxs) . (3.9)

These properties are easily transferred to the gyrogroup (Bn,⊕).

Corollary 1. The following equalities hold

(i) (sas)⊕ b = s(a⊕ (sbs))s and (ii) s(a⊕ b)s = (sas)⊕ (sbs) . (3.10)

The relation (sas) ⊕ (sbs) = s(a ⊕ b)s defines a homomorphism ofSpin(n) onto the gyrogroup (Bn,⊕).

From the decomposition (3.8) we can derive the Cartan or KAK de-composition of the group Spin+(1, n), whereK = Spin(n) and A = Spin(1, 1)is the subgroup of Lorentz boosts in the en−direction.

Lemma 4. Each a ∈ Bn can be written as a = srens, where r = |a| ∈ [0, 1[and s = ±s1 · · · sn−1 ∈ Spin(n) with

si = cosθi2+ ei+1ei sin

θi2, i = 1, . . . , n − 1, (3.11)

where 0 ≤ θ1 < 2π 0 ≤ θi < π, i = 2, . . . , n − 1.

This follows from the description of a ∈ Bn in spherical coordinatesusing the rotors (3.11). These rotors describe rotations in coordinate planes.For s = cos

(θ2

)+ eiej sin

(θ2

), i 6= j we have

sxs = (cos θ xi − sin θ xj)ei + (cos θ xj + sin θ xi)ej +n∑

k = 1k 6= i, j

xkek ,

which represents a rotation of angle θ in the ei+1, ei−plane. In general wehave sisj 6= sjsi, i 6= j. The order of the rotors is important since different

8 Milton Ferreira

choices lead to different systems of coordinates (c.f. formulae (6) and Lemma4.1 in [5]). Due to the relevance of the rotor sn−1 we shall denote θn−1 := φ.

Lemma 5. For a ∈ Bn as described in Lemma 4 it follows

ϕa(x) = ϕsrens(x) = sϕren(sxs)s.

Combining Lemma 5 with the rotation in decomposition (3.8) yieldsthe Cartan decomposition of Spin+(1, n).

For the construction of a theory of sections we need to consider somespecial gyro-subgroups.

Definition 3. Let (G,⊕) be a gyrogroup and H a non-empty subset of G.H is a gyro-subgroup of (G,⊕) if it is a gyrogroup with operation inducedfrom G and gyr[a, b]c ∈ Aut(H) for all a, b, c ∈ H.

Let ω ∈ Sn−1 and consider the hyperplane Hω = {x ∈ Rn : 〈ω, x〉 = 0},

the hyperdisc Dn−1ω = Hω ∩Bn and Lω = {x ∈ Bn : x = tω, −1 < t < 1}.

Proposition 4. For each ω ∈ Sn−1, the resulting sets Dn−1ω and Lω en-

dowed with the operation ⊕ are gyro-subgroups of (Bn,⊕). Moreover, (Lω,⊕)are groups.

The last property comes directly from Lemma 1. The operation ⊕ inLω corresponds to the Einstein velocity addition of parallel velocities in thespecial theory of relativity. The particular group (Len ,⊕) is isomorphic tothe Spin(1, 1) group.

4. Sections of the proper Lorentz group

We want to factorize the proper Lorentz group and to construct sectionsthat will allow us to define a conformal dilation operator for the unit sphere.First we will study the factorization of the gyrogroup (Bn,⊕) by the gyro-subgroup (Dn−1

en ,⊕), taking in consideration only left cosets.

Theorem 1. For each c ∈ Bn there exist unique a ∈ Dn−1en and b ∈ Len

such that c = b⊕ a.

Proof. Let c = (c1, . . . , cn) ∈ Bn be an arbitrary point. By Lemma 4we can write c = s∗c∗s∗, with s∗ = s1 . . . sn−2 ∈ Spin(n − 1), where thisrotation leaves the xn−axis invariant, and c∗ = (0, . . . , 0, c∗n−1, cn), wherec∗n−1 = r sinφ and cn = r cosφ, with r = |c| ∈ [0, 1[ and φ = arccos(cn) ∈[0, π]. If c∗n−1 = 0 then we take a = 0 and b = c∗. If c∗n−1 6= 0 then weconsider a = λen−1 and b = ten such that

λ =|c∗|2 − 1 +

√((cn + 1)2 + c∗

2

n−1)((cn − 1)2 + c∗2

n−1)

2c∗n−1

and t =cn

λc∗n−1 + 1.

(4.1)


We can see that −1 < λ, t < 1. Thus, a ∈ Dn−1en and b ∈ Len . Taking into

account that a ⊥ b, that is 〈a, b〉 = 0, we obtain

b⊕ a =

(0, . . . , 0,

λ(1− t2)

1 + λ2t2,t(1 + λ2)

1 + λ2t2

)= c∗.

Consider now a∗ = s∗as∗. Obviously, a∗ ∈ Dn−1en since the rotation induced

by s∗ leaves the xn−axis invariant. Then, by (3.10), we have

b⊕ (s∗as∗) = s∗((s∗bs∗)⊕ a)s∗ = s∗(b⊕ a)s∗ = s∗c∗s∗ = c,

which shows that c = b ⊕ a∗. Hence, the existence of the decomposition isproved.

To prove the uniqueness we suppose that there exist a, d ∈ Dn−1en and

b, f ∈ Len such that c = b⊕ a = f ⊕ d. Then a = (−b)⊕ (f ⊕ d), by (3.6).As b ⊥ d and f ⊥ d we have that a = ((−b)⊕ f)⊕ d, by Lemma 1. Since byhypothesis a, d ∈ Dn−1

en , (−b)⊕ f must be an element of Dn−1en . This is true

if and only if (−b)⊕ f = 0. This implies b = f and a = d⊕ 0 = d.

If we consider the subgroup Dn−1en and the relation R defined by

∀ c, d ∈ Bn, c R d ⇔ ∃ a ∈ Dn−1en : c = d⊕ a,

then R is a reflexive relation but it is not symmetric nor transitive becausethe operation ⊕ is not commutative nor associative. Thus R is not anequivalence relation.

However, an equivalence relation can be built using the following con-struction. We consider the sets Sb = {b⊕ a : a ∈ Dn−1

en }, with b ∈ Len .

Proposition 5. The family {Sb : b ∈ Len} is a disjoint partition of Bn.

Proof. We first prove that this family is indeed disjoint. Let b = t1enand c = t2en with t1 6= t2, and assume that Sb ∩ Sc 6= ∅. Then there existsf ∈ Bn such that f = b⊕a and f = c⊕d for some a, d ∈ Dn−1

en . By (3.6) and(3.5) we have a = (−b)⊕ (c⊕ d) = ((−b) ⊕ c) ⊕ (qdq), with q = 1+bc

|1+bc| . But

q = 1+bc|1+bc| =

1−t1t2|1−t1t2| = 1, hence a = ((−b) ⊕ c) ⊕ d. Then (−b) ⊕ c ∈ Dn−1

en

because a, d ∈ Dn−1en . Therefore, (−b) ⊕ c = 0 or either b = c. But this

contradicts our assumption.

Finally, by Theorem 1 we have that ∪b∈LenSb = Bn.

This partition induces an equivalence relation on Bn whose equivalenceclasses are the surfaces Sb. Two points c, d ∈ Bn are said to be in relation ifand only if there exists b ∈ Len such that c, d ∈ Sb, which means

c ∼l d ⇔ ∃b ∈ Len , ∃ a, f ∈ Dn−1en : c = b⊕ a and d = b⊕ f. (4.2)

10 Milton Ferreira

Thus, we have proved the isomorphism Bn/(Dn−1en ,∼l) ∼= Len . Given c ∈

Bn we denote by [c] its equivalence class on Bn/(Dn−1en ,∼l). A natural pro-

jection can be defined by π : Bn → Len , such that π(c) = b, by the uniquedecomposition of c ∈ Bn given in Theorem 1.

More general we have the isomorphisms Bn/(Dn−1ω ,∼l) ∼= Lω. We

will give a characterization of the surfaces Sb.

Proposition 6. For each b = ten ∈ Len the surface Sb is a revolutionsurface in turn of the xn-axis obtained by the intersection of Bn with thesphere orthogonal to ∂Bn, with center in the point C = (0, . . . , 0, 1+t

2

2t ) and

radius τ = 1−t22|t| .

Proof. Let b = ten ∈ Len , c = λen−1 ∈ Dn−1en and

P := b⊕ c =

(0, . . . , 0,

λ(1− t2)

1 + λ2t2,t(1 + λ2)

1 + λ2t2

).

Let Cb = {b ⊕ c : c = λen−1, 0 ≤ λ < 1}. Each a ∈ Dn−1en can be described

as a = s∗cs∗, for some c = λen−1 and s∗ = s1 · · · sn−2 ∈ Spin(n − 1). Then,by (3.10) we have

b⊕ (s∗cs∗) = s∗((s∗bs∗)⊕ c)s∗ = s∗(b⊕ c)s∗.

Thus, Sb is a surface of revolution obtained by revolution in turn of the xn-axis of the arc Cb. The last coordinate of the surface Sb gives the orientationof its concavity.

For all λ ∈ [0, 1[, we have ||P − C||2 = τ2, with C = (0, . . . , 0, 1+t2

2t )

and τ = 1−t22|t| , which means, the arc Cb lies on the sphere centered at C

and radius τ. Moreover, as t tends to zero the radius of the sphere tends toinfinity thus proving that the surface S0 coincides with the hyperdisc Dn−1

en .Each Sb is orthogonal to ∂B

n because ||C||2 = 1 + τ2.

Two spheres, S1 and S2, with centers m1 and m2 and radii τ1 andτ2, respectively, intersect orthogonally if and only if 〈m1 − y,m2 − y〉 =0, for all y ∈ S1 ∩ S2, or equivalently, if ||m1 −m2||2 = τ21 + τ22 .

FIGURE 1: Cut of the surfaces Sb, in the xn−1xn−plane.


The space Y = Bn/(Dn−1en ,∼l) becomes a homogeneous space endowed

with the mapping

h : Bn × Y → Y, h(c, [a]) = [c⊕ a]. (4.3)

We consider Len as the fundamental section σ0, which corresponds tothe Spin(1, 1) group. From Proposition 6 an entire class of sections σ :Bn/(Dn−1

en ,∼l) → Bn can be obtained from Len by considering

σ(ten) = ten ⊕ f(t)en−1 =

(0, . . . , 0,

f(t)(1− t2)

1 + (tf(t))2,t(1 + f(t)2)

1 + (tf(t))2

), (4.4)

where f :]−1, 1[→]−1, 1[ is the generating function of the section. Dependingon the properties of f we can have sections that are Borel maps and alsosmooth sections. If f ∈ Ck(] − 1, 1[), k ∈ N, then the section generates aCk-curve inside the unit ball. Two important families of sections considered

in this paper are σλ(t) =(0, . . . , 0, λ(1−t

2)1+t2λ2 ,

t(1+λ2)1+t2λ2

), t ∈]−1, 1[, for each λ ∈

[0, 1[ and σc(φ) = (0, . . . , 0, c sin φ, 0,− cos φ), φ ∈]0, π[, for each c ∈ [0, 1[.Each section σλ is generated by f(t) = λ, t ∈] − 1, 1[. For λ = 0 we obtainthe fundamental section. For c ∈]0, 1[ the generating function is obtainedby the intersection of the section σc with surfaces Sb and is given by

f(t) =

√t2−1+

√(1−t2)2+4c4t2

2t2c2, t ∈]− 1, 1[\{0}

c, t = 0.

Thus, we obtain a relation between the generating function and σc :

cosφ =

t(1+f(t)2)1+(tf(t))2 =

t2(2c2+1)−1+√

(1−t2)2+4c4t2

t(2c2+t2−1+√

(1−t2)2+4c4t2 ), t ∈]− 1, 1[\{0}

0, t = 0.

For c = 0, we obtain again the fundamental section.

Until now we only considered the factorization of Bn. In order to in-corporate rotations in our scheme we now extend the equivalence relation(4.2) to the group Spin+(1, n) according to the group operation (3.8). It iseasy to see that the direct product {1}×Dn−1

en is a gyrogroup, where 1 is theidentity rotation in Spin(n). Our goal is to define an equivalence relation ∼∗

l

on Spin(n)×Bn, which is an extension of the equivalence relation ∼l on Bn,

such that the homogeneous space X = (Spin(n)×Bn)/({1} ×Dn−1en ,∼∗

l ) isisomorphic as a set to Spin(n)× Len . For (s1, c), (s2, d) ∈ Spin(n) × Bn we

12 Milton Ferreira

define the equivalence relation ∼∗l by

(s1, c) ∼∗l (s2, d) ⇔ ∃ s3 ∈ Spin(n), ∃ b ∈ Len , ∃ a, f ∈ Dn−1

en :

(s1, c) =

(s3

1− [(−a)⊕ (b⊕ a)]a

|1− [(−a)⊕ (b⊕ a)]a| , (−a)⊕ (b⊕ a)

)× (e, a)

and (4.5)

(s2, d) =

(s3

1− [(−f)⊕ (b⊕ f)]f

|1− [(−f)⊕ (b⊕ f)]f | , (−f)⊕ (b⊕ f)

)× (e, f).

The equivalence relation (4.5) reduces to

(s1, c) ∼∗l (s2, d) ⇔ s1 = s2 ∧ (c = b⊕ a ∧ d = b⊕ f).

It is easy to see that the equivalence class associated to (s1, c) is equal to{s1} × [c], with [c] = Sb ∈ Bn/(Dn−1

en ,∼l), and the quotient space is X ={{s} × Sb : s ∈ Spin(n), b ∈ Len} ∼= Spin(n) × Len . The group Spin+(1, n)acts on X according to (3.8) by the mapping

g : Spin+(1, n) ×X → X

((s1, a), (s2, [c])) 7→(s1s2

1− s2as2c

|1− s2as2c|, [c⊕ (s2as2)]

).

The space X will be the underlying homogeneous space for the con-struction of spherical continuous wavelet transforms.

5. Wavelet theory for homogeneous spaces

We will present the theory developed for homogeneous spaces ([1], [2]) andapplied e.g. in [7], and [8]. Let G be a locally compact group with left Haarmeasure µ and let H be a separable Hilbert space. A strongly continuousunitary representation of G on H is a mapping U from G into the unitaryoperators on H for which U(g1g2) = U(g1)U(g2) for all g1, g2 ∈ G and themapping g 7→ U(g)f is continuous for all f ∈ H. Further, U is square-integrable if there exists ψ ∈ H\{0} such that

∫

G| 〈ψ,U(g)ψ〉H |2dµ(g) <∞.

For the classical integral transforms like the short time Fourier transformand the wavelet transform related to the reduced Weyl-Heisenberg groupand the affine group, respectively, the representations in question are squareintegrable. However, for integral transforms related to group representations


on L2 spaces on manifolds, for example on the sphere, square integrabilityfails to hold. In other words, the corresponding group is too large.

A way to overcome this fact, is to make the group G smaller, i.e. tofactor out a suitable closed subgroup H. In this way we restrict the repre-sentation to a quotient G/H. Since the representation is not defined directlyon G/H (unless H is a normal subgroup) we need to introduce a sectionσ : G/H → G which assigns to each coset a point lying in it. In otherwords, if π : G→ G/H denotes the canonical projection then π ◦ σ = id. Ingeneral the section σ cannot be chosen continuous but it is always possibleto chose it measurable or continuous on some dense open subset of G/H.

Suppose µ is a Radon measure on G/H and define the translate µx byµx(E) = µ(xE) for a measurable set E and x ∈ G. If all measures µx, x ∈ Gare equivalent (i.e. they have the same null sets) then µ is called a quasi-invariant measure. If there exists a continuous function λ : G × G/H →(0,∞) such that dµx(p) = λ(x, p)dµ(p) for all x ∈ G, p ∈ G/H then µ iscalled strongly quasi-invariant.

So given a group G, a closed subgroup H, a section σ, a quasi-invariantmeasure µ and a representation U of G in H one says that the representationis square-integrable modulo (H,σ) if there exists a non-zero ψ ∈ H such that

∫

G/H| 〈f, U(σ(p))ψ〉 |2dµ(p) <∞ for all f ∈ H. (5.1)

(We changed the notation of [8] replacing σ(p)−1 by σ(p) in order to be morecompatible with our example.)

We may define an operator Aσ (dependent on σ and ψ) weakly by

Aσf =

∫

G/H〈f, U(σ(p))ψ〉U(σ(p))ψ dµ(p), (5.2)

in the sense that

〈Aσf , g〉 =∫

G/H〈f, U(σ(p))ψ〉〈U(σ(p))ψ, g〉 dµ(p) for all f, g ∈ H.

By (5.1), the operator Aσ is clearly positive and bounded. If Aσ is also acontinuous frame, i.e., if there exists constants 0 < A ≤ B <∞ such that

A||f ||2 ≤∫

G/H| 〈f, U(σ(p))ψ〉 |2dµ(p) ≤ B||f ||2, for all f ∈ H,

then the operator Aσ is invertible with a continuous inverse and ψ is calledadmissible. If furthermore Aσ is a constant multiple of the identity then ψis called strictly admissible. The wavelet transform is defined by

Vψf(p) := 〈f, U(σ(p))ψ〉 , p ∈ G/H.

14 Milton Ferreira

In the case that Aσ is a multiple of the identity one uses Vψ to define thereproducing kernel. In the general case we define another transform

Wψf(p) := Vψ(A−1σ f)(p) =

⟨A−1σ f, U(σ(p))ψ

⟩, p ∈ G/H.

We have for all f, g ∈ H

〈f, g〉 =⟨AσA

−1σ f, g

⟩

=

∫

G/H

⟨A−1σ f , U(σ(p))ψ

⟩〈U(σ(p))ψ, g〉 dµ(p). (5.3)

Setting g = A−1σ U(σ(q))ψ, q ∈ G/H and using the self-adjointness of Aσ we

obtain the reproducing formula

Wψf(q) =⟨f,A−1

σ ψ⟩

=

∫

G/H

⟨A−1σ f, U(σ(p))ψ

⟩ ⟨U(σ(p))ψ,A−1

σ U(σ(q))ψ⟩dµ(q)

=

∫

G/HWψf(p)R(q, p) dµ(p)

where Rψ(q, p) :=⟨U(σ(p))ψ,A−1

σ U(σ(q))ψ⟩= Wψ(U(σ(p)))(q). Clearly,

Rψ(q, p) = Rψ(p, q) for all p, q ∈ G/H. Furthermore, the map Wψ is anisomorphism of H onto the reproducing kernel Hilbert space

M2 := {F ∈ L2(G/H) : 〈F,Rψ(q, ·)〉 = F (q) a.e.}.

If we introduce the mapping Wψ : L2(G/H) → H defined by WψF =∫G/H F (p)U(σ(p))ψ dµ(p) then it holds f = WψWψf, for all f ∈ H in a

weak sense by formula (5.3).

6. Lorentz Coherent States as Wavelets on Sn−1

The theory in the previous section depends only on the homogeneous space,on the choice of a section and a quasi-invariant measure on the homoge-neous space, and on the representation of the group. Therefore, the generalapproach also works in our case since we can obtain these terms in our case.We will still use the term square-integrability modulo (H,σ), where H de-notes only a gyro-subgroup because our homogeneous space X results fromthe factorization of a group by a gyro-subgroup.

For the construction of a spherical continuous wavelet transform weneed to define motions and dilations on Sn−1. Motions are given by theaction of elements of Spin(n) and dilations will arrise from Mobius transfor-mations ϕa. In [5] we studied the influence of the parameter a on spherical


caps centered at the North Pole. We present here the most important resultsin order to proceed with our construction. For more details see [5].

We denote by Uh = {x ∈ Sn−1 : xn ≥ h}, for some h ∈] − 1, 1[, anarbitrary spherical cap and by Uh,a the cap ϕa(Uh). The center of cap Uh,aon the unit sphere is the point

C =

(2a1(an − h)√

kh(a), . . . ,

2an−1(an − h)√kh(a)

,1− |a|2 + 2an(an − h)√

kh(a)

), (6.1)

where kh(a) := 4(an − h)2(|a|2 − a2n) + (1− |a|2 + 2an(an − h))2.The distance between the center of the cap Uh,a and the hyperplane

that supports the cap is given by

dh(a) = 1 +2an − h(1 + |a|2)√

kh(a). (6.2)

Therefore, by expressing the point a in spherical coordinates, a local dila-tion on the unit sphere depends on two parameters r ∈ [0, 1[ and φ ∈ [0, π].Moreover we can separate points a ∈ Bn in two different regions: a dila-tion and a contraction regions. These regions are separated by a revolutionsurface obtained by the revolution around the xn−axis of the arc defined by

−→γ (r) = (r (1− (hr)2)1/2, 0, . . . , 0, r2 h), r ∈ [0, 1[. (6.3)

The cap Uh,a is characterized by the existence of an attractor point A(see Fig. 2), which is defined as the image of the North Pole under ϕa andis given by

A =

(2a1(an − 1)

1 + |a|2 − 2an, . . . ,

2an−1(an − 1)

1 + |a|2 − 2an,1− |a|2 + 2an(an − 1)

1 + |a|2 − 2an

). (6.4)

By (6.1) we see that the cap Uh,a is not centered at the North Poleif a ∈ Bn\Len or an 6= h. Applying a convenient rotation to each capUh,a we can center all spherical caps in an arbitrary desired point of thesphere. For instance, in the case n = 3 (the sphere S2), if we considersh,a = cos β/2 + ω sin β/2 ∈ Spin(3) with

ω =

[− ǫ1a2√

a21+a22

, ǫ1a1√a21+a

22

, 0

]T, cos β = 1−|a|2+2a3(a3−h)√

kh(a), if a ∈ B3\Le3

ω = e3 , β = 0, if a ∈ Le3

(6.5)

where ǫ1 = +1 if h ≥ a3 and ǫ1 = −1 if h < a3, then the set {saUh,asa : a ∈Bn} stands for a family of caps centered at the North Pole.

The sections σλ or σc give rise to conformal dilation operators on Sn−1

with different properties. For λ = 0 or c = 0 we obtain the operator ϕten

16 Milton Ferreira

1 2 3

FIGURE 2: Spherical caps for h = cos(π/6) =√3/2 and different values

of a : 1 - a = (0, 0, 0) (U√3/2), 2 - a = (1/8,−

√3/8,

√3/4), 3 - a =

(1/4, 1/8,−1/4).

(with t = − cosφ in the second case) that corresponds to a pure dilation onSn−1. For λ, c ∈]0, 1[, the operators ϕσλ and ϕσc correspond to anisotropicdilations.

For f ∈ L2(Sn−1) the rotation and spherical dilation operators aredefined by

Rsf(x) = f(sxs) and Daf(x) =

(1− |a|2|1− ax|2

)n−12

f(ϕ−a(x)), (6.6)

where s ∈ Spin(n), a ∈ Bn, and(

1−|a|2|1−ax|2

)n−1is the Jacobian of ϕ−a(x) on

Sn−1. From these operators we construct the representation U of Spin+(1, n)in L2(Sn−1, dS) defined by

U(s, a)f(x) = RsDaf(x) =

(1− |a|2

|1− asxs|2)n−1

2

f(ϕ−a(sxs)). (6.7)

Proposition 7. U(s, a) is a unitary representation of the group Spin+(1, n)in L2(Sn−1).

Proof. We are going to prove that U(s, a) is a homomorphism of the groupSpin+(1, n) onto the space of linear maps from the Hilbert space L2(Sn−1)onto itself, with respect to the group operation (3.8).

On the one hand we have

U(s1, a)(U(s2, b)f(x)) =

=

„

1− |a|2

|1− as1xs1|21− |b|2

|1− bs2ϕ−a(s1xs1)s2|2

«

n−12

f(ϕ−b(s2ϕ−a(s1xs1)s2))

=

„

(1− |a|2)(1− |b|2)

|1− s2bs2a− (a+ s2bs2)s1xs1|2

«

n−12

f(ϕ−(b⊕(s2as2))(s3xs3)),


with s3 = s1s21−s2as2b|1−s2as2b| .

We observe that

|1− bs2ϕ−a(s1xs1)s2|2 = |1− s2bs2(s1xs1 + a)(1− as1xs1)

−1|2

= |1− as1xs1|−2|1− as1xs1 − s2bs2(s1xs1 + a)|2

= |1− as1xs1|−2|1− s2bs2a− (a+ s2bs2)s1xs1|

2.

By (3.9), (3.2) and (3.4) it follows

ϕ−b(s2ϕ−a(s1xs1))s2) = ϕ−b(ϕ−s2as2)(s2 s1xs1s2)

= q1ϕ−(b+s2as2)(1−s2as2b)−1(s1s2xs1s2)q1

= ϕ−q1((s2as2)⊕b)q1 (q1s1s2xs1s2q1)

= ϕ−(b⊕(s2as2))(s3xs3),

where q1 =1−bs2as2|1−bs2as2| .

On the other hand we have

U(s3, b⊕ (s2as2))f(x) =

„

1− |b ⊕ (s2as2)|2

|1− (b⊕ (s2as2))s3xs3|2

«

n−12

f(ϕ−(b⊕(s2as2))(s3xs3))

=

„

(1− |a|2)(1− |b|2)

|1− s2bs2a− (a+ s2bs2)s1xs1|2

«

n−12

f(ϕ−(b⊕(s2as2))(s3xs3)).

We note that

1− |b⊕ (s2as2)|2 = 1− |ϕ−b(s2as2)|

2 =(1− |a|2)(1− |b|2)

|1− bs2as2|2

and (by (3.4))

|1− (b⊕ (s2as2))s3xs3|2 = |1− q2(b⊕ (s2as2))q2 s2 s1xs1s2|

2

= |1− (s2as2 ⊕ b) s2 s1xs1s2|2

= |1− bs2as2|−2|1− bs2as2 − (s2as2 + b)s2 s1xs1s2|

2

= |1− bs2as2|−2|s2(1− s2bs2a)− s2(a+ s2bs2) s1xs1|

2|s2|2

= |1− bs2as2|−2|1− s2bs2a− (a+ s2bs2) s1xs1|

2,

with q2 =1−s2as2b|1−s2as2b| .

Thus, U(s1, a)(U(s2, b)f(x)) = U(s3, b ⊕ (s2as2))f(x), i.e. U(s, a) is arepresentation of Spin+(1, n) on L2(Sn−1).

As the operators (6.6) are unitary then the representation U(s, a) isunitary, i.e. ||U(s, a)f ||L2(Sn−1) = ||f ||L2(Sn−1).

This representation can be decomposed via the Cartan decompositionof Spin+(1, n). It belongs to the principal series of SO0(1, n), being irre-ducible on the space L2(Sn−1).

Following the general approach of [1] we will build a system of co-herent states for the Lorentz group Spin+(1, n), indexed by points of thehomogeneous space X. We will consider the parameter space W = (s, σ),where s ∈ Spin(n) and σ = σλ or σ = σc, which corresponds to a sec-tion on X. Moreover we will consider the restriction of the representation

18 Milton Ferreira

U to W. For the fundamental section Len the representation U(s, σ0) coin-cides with the representation used in [4] (see [5]). Now, we have to con-struct an quasi-invariant measure on X. First we observe that the measure

dµ(σ0) = 2(1−t)n−2

(1+t)n dt is a quasi-invariant measure on Y = Bn/(Dn−1en ,∼l).

For all a ∈ Bn and b ∈ Len such that a = s∗a∗s∗ we have that −a ⊕b = (s∗(−a∗)s∗) ⊕ b = s∗(−a∗ ⊕ (s∗bs∗))s∗ = s∗(−a∗ ⊕ b)s∗. Thus, [−a ⊕b] = [−a∗ ⊕ b]. This means that we only need to consider the action ofa = (0, . . . , 0, an−1, an) on b = ten ∈ Len . Let d = −a ⊕ b = ϕa(b) =(0, . . . , 0, −(1+t2−2ant)an−1

1−2ant+|a|2t2 , (1−|a|2)t−(1+t2−2ant)an1−2ant+|a|2t2

). Applying the projection

formulas (4.1) we obtain the new class of equivalence, given by

t∗ = ga(t) =−2an(1 + t2) + 2(1− a2n−1 + a2n)t√C1(t)C2(t) + (1 + |a|2)(1 + t2)− 4tan

,

with C1(t) := (1− t)2a2n−1 + (1 + t)2(1− an)2 and C2(t) := (1 + t)2a2n−1 +

(1− t)2(1 + an)2. Differentiating with respect to t we obtain

g′a(t) =2(1 − t2)(1 + a2n−1 − a2n)(1− |a|2)

C1(t)C2(t) + ((1 + |a|2)(1 + t2)− 4ant)√C1(t)C2(t)

.

Therefore, the Radon-Nikodym derivative of dµ([−a⊕ b]) with respectto dµ(b) is given by:

χ(a, b) =dµ([−a⊕ b])

dµ(b)=

(1− ga(t))n−2(1 + t)n

(1 + ga(t))n(1− t)n−2g′a(t).

Since for each a we have that g′a(t) > 0, for all t ∈]− 1, 1[ we conclude thatχ(a, b) ∈ R

+, for all a ∈ Bn and b ∈ Len , thus proving that the measuredµ(b) is quasi-invariant. Moreover this measure is equivalent to the measuredµ(u) = du

un , by means of the bijection given by t = u−1u+1 (u ∈ R

+ and

t ∈]− 1, 1[ ). For f ∈ L1(Len) we have∫

Len

f([a⊕ b])dµ(b) =

∫

Len

f(b)dµ([−a⊕ b]) =

∫

Len

f(b)χ(a, b)dµ(b).

For the special case of a = (0, . . . , 0, an) ∈ Len we obtain χ(a, b) =(1+an1−an

)n−1.

The behavior of χ(a, b) depends on the dimension. For n = 3 the functionχ(a, b) is bounded in the variable b, i.e. for each a ∈ Bn there exists a

constant M(a) =(1−|a|2)(1+a2n−1−a2n)

(1−an)4 ∈ R+ such that χ(a, b) ≤ M(a), for all

b ∈ Len , since χ(a, b) is an increasing function in the variable b, for eacha 6= 0. Thus, dµ([−a⊕ b]) ≤M(a) dµ(b). For n > 3 this estimate is not validsince the function χ(a, b) is not bounded in the variable b.

A quasi-invariant measure for X can now be given by dµ(s, σ0) =2(1−t)n−2

(1+t)n dt dµ(s), associated to the section (s, σ0), where dµ(s) is the in-

variant measure on Spin(n).


For instance, the measure dµ(s, σλ) = χ(σλ, b)dµ(σ0) is the standardquasi-invariant measure for the section W = (s, σλ). (see [1]). We knowthat the quasi-invariant measure on a section is unique (up to equivalence),that is, if µ1 and µ2 are quasi-invariant measures on X then there is aBorel function f : X → R

+, f(x) > 0, for all x ∈ X, such that dµ1(x) =f(x)dµ2(x), for all x ∈ X. We will consider the family of quasi-invariantmeasures

dµ(σλ(t)) =2(1 + λ(1−t2)

1+t2λ2− t(1+λ2)

1+t2λ2

)n−2

(1 + λ(1−t2)

1+t2λ2+ t(1+λ2)

1+t2λ2

)n dt. (6.2)

For λ = 0 we obtain the quasi-invariant measure dµ(σ0(t)) =2(1−t)n−2

(1+t)n dt on

the fundamental section. Moreover, for each λ ∈]0, 1[ it holds dµ(σλ(t)) ≤dµ(σ0(t)), for all t ∈]− 1, 1[.

Let N(n, k) be the number of all linearly independent homogeneousharmonic polynomials of degree k in n variables and {H

(i)k , i = 1, . . . , N(n, k)}∞k=0

be an orthonormal basis of spherical harmonics, i.e. < H(i)k ,H

(j)l >L2=

δk,lδi,j. Thus, a function f ∈ L2(Sn−1) has a Fourier expansion

f(x) =∞∑

k=0

N(n,k)∑

i=1

a(i)k H

(i)k (x), (6.3)

where a(i)k :=< H

(i)k , f >L2(Sn−1) are the Fourier coefficients of f .

Theorem 2. The representation U given in (6.7) is square integrablemodulo ({1} × Dn−1

en ,∼∗l ) and the section (s, σλ), if there exists a nonzero

admissible vector ψ ∈ L2(Sn−1, dS) satisfying

1

N(n, k)

N(n,k)∑

i=1

∫ 1

−1|a(i)k (t)|2 dµ(σλ(t)) <∞, (6.4)

uniformly in k, where a(i)k (t) =< H

(i)k ,DSn−1

σλψ >L2(Sn−1) .

If condition (6.4) is satisfied for some ψ ∈ L2(Sn−1) we obtain a con-tinuous family of coherent states indexed by points (s, σλ). An admissibilitycondition can also be stated for the sections (s, σc).

7. The case of the sphere S2

The relationship between SL(2,C) and SO0(1, 3) is well known: the groupSL(2,C) being isomorphic to the group Spin(1, 3) is a (simply connected)double-cover of SO0(1, 3).

20 Milton Ferreira

Considering S2 the Riemann sphere and identifying the tangent planeat the North Pole with the complex plane we denote by Φ the stereographicprojection map Φ : S2 → C ∪ {∞} given by

Φ(x1, x2, x3) =2x1 + 2x2i

1 + x3, (x1, x2, x3) ∈ S2\{(0, 0,−1)},

and Φ(0, 0,−1) = ∞. The inverse mapping Φ−1 : C∪ {∞} → S2 is given by

Φ−1(y1 + y2i) =

(4y1

4 + r2,

4y24 + r2

,4− r2

4 + r2

), Φ−1(∞) = (0, 0,−1),

with r2 = y21 + y22.As Mobius transformations map spheres into spheres and the stereo-

graphic projection is a conformal mapping, the Mobius transformation ϕa(acting on S2) can be identified with a Mobius transformation on the com-pactified complex plane C, i.e. an element of SL(2,C).

Theorem 3. For each a = (a1, a2, a3) ∈ B3 the Mobius transformation ϕaacting on S2 can be described as an element of SL(2,C) by the matrix

1+a3√1−|a|2

−2(a1+a2i)√1−|a|2

−a1+a2i2√

1−|a|21−a3√1−|a|2

. (7.1)

Sketch of the Proof:Case 1: a1 6= 0 and a2 6= 0 : Let T denotes the Mobius transformationon C that corresponds to ϕa on S2. Let T (2) = Φ(ϕa(1, 0, 0)), T (0) =Φ(ϕa(0, 0,−1)), and T (∞) = Φ(ϕa(0, 0, 1)). As T is invariant under the crossratio it holds (T (z), T (∞), T (2), T (0)) = (z,∞, 2, 0). The left hand-side isequal to (z − 2)/z. Solving this cross ratio we will find by straightforwardcomputations

T (z) =(2 + 2a3)z − 4a1 − 4a2i

(−a1 + a2i)z + 2− 2a3.

After normalization we obtain the desired result.Case 2: a1 = a2 = 0 (Spin(1, 1)−case): Let T (2) = Φ(ϕa(1, 0, 0)),

T (0) = Φ(ϕa(0, 0,−1)) and T (−2) = Φ(ϕa(−1, 0, 0)). Solving the cross ratiorelation (T (z), T (−2), T (2), T (0)) = (z,−2, 2, 0) we will find by straightfor-ward computations the following normalized matrix of SL(2,C)

√

1+a31−a3 0

0√

1−a31+a3

.


The Iwasawa decomposition of the group SL(2,C) yields the decom-position SL(2,C) = KAN, where K = SU(2), is the maximal compactsubgroup, A is abelian and N is nilpotent. The Iwasawa decomposition ofa generic element of SL(2,C) reads

(u vw z

)=

(α β

−β α

)(δ−1/2 0

0 δ1/2

)(1 ξ0 1

)(7.2)

where α, β, ξ ∈ C and δ ∈ R+, uz − vw = 1 and

δ = (|u|2 + |w|2)−1, α = uδ1/2, β = −wδ1/2, ξ = u−1(v + wδ).

The Iwasawa decomposition of the matrix (7.1) yields the parameters

α = 1−a3√1+|a|2−2a3

, β = a1+a2i√1+|a|2−2a3

, δ = 1−|a|21+|a|2−2a3

, ξ = −2(a1+a2i)1+|a|2−2a3

.

For example, for the sections σλ we obtain the parameters

αλ = 1−tλ2√(1+λ2)(1+t2λ2)

, βλ = λ(1+t)√(1+λ2)(1+t2λ2)

i,

δλ = (1+t)(1−λ2)(1−t)(1+λ2) , ξλ = 2(1+t)λ

(t−1)(1+λ2)i,

while for the sections σc we obtain the parameters

αc =1+cos φ√

(1+cos φ)2+c2 sin2 φ, βc =

c sinφ√(1+cos φ)2+c2 sin2 φ

i,

δc =(1−c2) sin2 φ

(1+cos φ)2+c2 sin2 φ, ξc =

−2c sinφ(1+cosφ)2+c2 sin2 φ

i .

Geometrically the sections σλ and σc generate similar curves inside theunit ball. However, there is a difference between both families of sections.For λ ∈ [0, 1[ it holds 0 < δλ < ∞, while for c ∈]0, 1[ it holds 0 < δc <1−c2c2

. This happens since the respective generating functions have differentbehavior at the endpoints of the interval ] − 1, 1[. Thus, for c ∈]0, 1[ theoperator Dδc cannot be considered as a dilation operator.

We need another idea to perform the stereographic projection of oursections. The idea is to see our sections as a Mobius deformation of thefundamental section and to compare the new sections with the fundamentalone. In Section 6 we observed that if Uh is a spherical cap centered at theNorth Pole then the cap Uh,a is not centered at the North Pole if a ∈ Bn\Lenor an 6= h. In order to obtain a dilation operator on the tangent plane onehas to move the tangent plane to the center of the cap Uh,a and then toperform the stereographic projection. This is equivalent to rotate all capsto the North Pole and then to perform the stereographic projection. This isexactly our local dilation around the North Pole defined in [5].

22 Milton Ferreira

Let Uh,a be the spherical cap Uh,a rotated to the North Pole, i.e. Uh,a =sh,aUh,ash,a, for some rotation sh,a ∈ Spin(3) (see (6.5)).

We want to give a description of the geometry of the disc Φ(Uh,a) on thetangent plane. Since we are dealing with conformal mappings it is enough tostudy the mapping of the boundary and the mapping of the attractor pointto get the mapping property of any point in the disc Φ(Uh,a). We rememberthat a Mobius transformation on an arbitrary disc Dr(0) (centered at theorigin and radius r) is given by rϕρ(z/r), z ∈ Dr(0), where ϕρ(z) is a Mobiustransformation on the unit disc defined by ϕρ(z) = z+ρ

1+ρz , with ρ ∈ D1(0).Therefore we have the following lemma.

Lemma 6. Let a = (0, a2, a3) ∈ B3 and −1 < h < 1. The stereographicprojection of the action sh,a ϕa(Uh) sh,a onto the tangent plane of S2 is givenby

Φ(sh,a ϕa(x) sh,a) = ϕρh(a)(δh(a)Φ(x)), x ∈ Uh, (7.3)

where δh(a) =(1−|a|2)(1+h)

(1+|a|2)h−2a3+√kh(a)

, with kh(a) := 4(a3 − h)2a22 + (1− |a|2 +

2a3(a3 − h))2, and ϕρh(a)(z) = r2z+ρh(a)r2r2+ρh(a)z

, is a Mobius transformation on

Dr2(0), with r2 =2δh(a)

√1−h2

1+h , and ρh(a) = − 2(1−h)a2((1+|a|2)h−2a3+√kh(a))√

1−h2(√kh(a)(1+|a|2−2a3)+Ch(a))

,

with Ch(a) := kh(a)− 2(1− h)((1 + |a|2 − 2ha3)a3 − 2ha22).

Proof. Let P1 = (0,√1− h2, h) and P2 = (0,

√1− h22, h2), with h2 =

1 − dh(a) = −2a3+(1+|a|2)h√kh(a)

, be two points on the spherical support of the

caps Uh and Uh,a, respectively. Then Φ(P1) =(0, 2

√1−h21+h

)and Φ(P2) =

(0, 2(1−|a|2)

√1−h2

(1+|a|2)h−2a3+√kh(a)

). Thus, the radius of the discs Φ(Uh) and Φ(Uh,a)

are given by r1 = 2√1−h21+h and r2 = 2(1−|a|2)

√1−h2

(1+|a|2)h−2a3+√kh(a)

, respectively. The

ratio between these radius gives the parameter δh(a), i.e. δh(a) = r2r1

=(1−|a|2)(1+h)

(1+|a|2)h−2a3+√kh(a)

.

Let d be the spherical distance between the center of the cap Uh,a and

its attractor point A = ϕa(e3) =(0, 2a2(a3−1)

1+|a|2−2a3, 1−|a|2+2a3(a3−1)

1+|a|2−2a3

), which is

given by

cos(d) =a3(a

33 − 2(1 + h)(1 + |a|2) + 2(1 + a22 + 2h)a3) + 1 + a42 + (4h− 2)a22√

kh(a)(1 + |a|2 − 2a3).

Since this distance is invariant under rotations we can easily obtain thecoordinates of the attractor point A on the cap Uh,a, which are given by

A = (0, ǫ√

1− cos2(d), cos(d)) with ǫ = −1 if a2 > 0 or ǫ = +1 if a2 ≤ 0.Note that if a2 > 0 then the second component of A is negative, otherwise it


is non-negative. The stereographic projection of A yields the point Φ(A) =(0, ǫ

2√

1−cos2(d)

1+cos(d)

). Thus, we obtain the parameter ρh(a) = ǫ

2√

1−cos2(d)

r2(1+cos(d)) i,

(after rescaling Φ(A) by the radius r2). By straightforward computationswe obtain

ρh(a) = ǫ2(1 − h)|a2|((1 + |a|2)h− 2a3 +

√kh(a))√

1− h2((1 + |a|2 − 2a3)√kh(a) +Ch(a))

i,

= − 2(1 − h)a2((1 + |a|2)h− 2a3 +√kh(a))√

1− h2((1 + |a|2 − 2a3)√kh(a) + Ch(a))

i,

with Ch(a) := kh(a)− 2(1 − h)((1 + |a|2 − 2ha3)a3 − 2ha22).

Figure 3 exemplifies Lemma 6 in a particular case to show that theintertwining relation (7.3) holds pointwise.

FIGURE 3: R.h.s.: Φ(sh,a ϕa(Uh)sh,a), L.h.s.: ϕρh(a)(δh(a)Φ(Uh)), for h =

cos(π/6) =√3/2 and a = (0, 1/2 sin(3/4π), cos(3/4π)).

We can easily generalize relation (7.3) for an arbitrary point a ∈ B3

since only the position of the attractor point ρh(a) is affected by a rotation.Indeed, considering a = s∗a∗s∗, with a∗ = (0, a2, a3) and s∗ ∈ Spin(2) weobtain the relation

Φ(sh,a ϕa(Uh) sh,a) = ϕs∗ρh(a∗)s∗(δh(a∗)Φ(Uh)).

If a = ten ∈ Len the parameter δh(a) is independent of h since δ(ten) =1+t1−t and ρh(ten) = 0, which shows that the anisotropic effect disappears ifwe consider the fundamental section.

Figure 4 shows the behavior of |ρh(a)| for h = 0 and a = (0, r sinφ, r cosφ),r ∈ [0, 1[ and φ ∈ [0, π].

From now on we will use f ∈ L2(S2) and F ∈ L2(R2) to distinguishfunctions from different spaces. We will also sometimes make use of the

24 Milton Ferreira

FIGURE 4: Behavior of |ρh(r, φ)| for h = 0.

identification of R2 with the complex plane C.

Lemma 7. The map Θ : L2(S2, dS) → L2(R2, rdr dθ) defined by

Θ : f(θ, φ) 7→ F (θ, r) =4

4 + r2f(θ, 2 arctan(r/2)) (7.3)

is a unitary map. In cartesian coordinates the map Θ reads as

Θ : f(x1, x2, x3) 7→ F (y1, y2) =4

4 + y21 + y22f(Φ−1(y1, y2)). (7.4)

As a consequence of Lemma 6 we obtain the following Lemma.

Lemma 8. For each h ∈]− 1, 1[ and ψ ∈ L2(S2) such that supp(ψ) ⊂ Uhwe have the intertwining relation

ΘRsh,aDaψ =Mρh(a)Dδh(a)Θψ (7.5)

where Mρh(a), and Dδh(a) are the unitary operators on L2(C) defined by

Dδh(a)F (z) = 1δh(a)

F(

zδh,a

), andMρh(a)F (z) =

r22(1−|ρh(a)|2)|r2−ρh(a)z|2

F(ϕ−ρh(a)(z)

),

z ∈ C.

Proof. By the definition of our operators we have

ΘRsh,aDaψ(y) =4

4 + r21− |a|2

|1− ash,aΦ−1(y)sh,a|2ψ(ϕ−a(sh,aΦ

−1(y)sh,a))

and

Mρh(a)Dδh(a)Θψ(y) =

1

δh(a)

r22(1 − |ρh(a)|2)|r2 + ρh(a)y|2

4

4 +

(ϕ

[1]

−ρh(a)(y)

δh(a)

)2

+

(ϕ

[2]

−ρh(a)(y)

δh(a)

)2

ψ

(Φ−1

(ϕ−ρh(a)(y)

δh(a)

)),


where ϕ[1]−ρh(a)(y) and ϕ

[2]−ρh(a)(y) are the components of ϕ−ρh(a). First we

prove that

ϕ−a(sh,aΦ−1(y)sh,a) = Φ−1

(ϕ−ρh(a)(y)

δh(a)

). (7.5)

If ϕ−a(sh,aΦ−1(y)sh,a) = x ∈ S2 then y = Φ(sh,a ϕa(x) sh,a). Moreover, if

Φ−1(ϕ−ρh(a)(y)

δh(a)

)= x then y = ϕρh(a)(δh(a)Φ(x)). Thus, by (7.3) we conclude

that (7.5) is true.Second, by straightforward computations we can check that

4

4 + r21− |a|2

|1− ash,aΦ−1(y)sh,a|2=

1

δh(a)

r22(1− |ρh(a)|2)

|r2 + ρh(a)y|24

4 +

ϕ[1]−ρh(a)

(y)

δh(a)

!2

+

ϕ[2]−ρh(a)

(y)

δh(a)

!2,

for every y ∈ Dr2(0).

Corollary 2. For a = te3 ∈ Le3 (restriction to Spin(1, 1)) we obtain theintertwining relation

ΘDaψ = D1+t1−tΘψ. (7.6)

Proof. In the case of Spin(1, 1) we have sh,a = 1, ρh(a) = 0 and δh(a) =1+t1−t . Thus,

ΘDaψ(y) =4

4 + r21− t2

|1− te3Φ−1(y)|2 ψ(ϕ−te3(Φ−1(y)))

=4(1 − t2)

4(1 + t)2 + r2(1− t)2ψ(ϕ−te3(Φ

−1(y)))

and

D1+t1−tΘψ(y) =

11+t1−t

4

4 +(r2(1−t)(1+t)

)2 ψ(Φ−1

(y

1+t1−t

)),

=4(1 − t2)

4(1 + t)2 + r2(1− t)2ψ

(Φ−1

((1− t)y

1 + t

)).

For ρ ∈ D1(0) and τ1, τ2 > 0 we define the following unitary operator

M τ1,τ2ρ : L2(Dτ1(0)) → L2(Dτ2(0))

M τ1,τ2ρ F (z) =

τ1τ2(1− |a|2)|τ2 − ρz|2 F

(τ1ϕ−ρ

(z

τ2

)),

where ϕρ(z) =z+ρ1+ρz is a Mobius transformation on the unit disc D1(0).

26 Milton Ferreira

Thus, we have the following relations:

Mρh(a)Dδh(a)Θψ = M τ2,τ2

ρh(a)Dδh(a)Θψ (7.5)

= Dr2M1,1ρh(a)

D1/r2Dδh(a)Θψ (7.6)

= Dr2M1,1ρh(a)

D1/r1Θψ (7.7)

= Dr2M r2,1ρh(a)

Dδh(a)Θψ (7.8)

= Dr2M r1,1ρh(a)

Θψ, (7.9)

where r1 and r2 are the radius of the discs Φ(Uh) and Φ(Uh,a), respectively.The operator Mρh(a) produces the anisotropic effect in the geometry of

the disc Dr2(0). Being Mρh(a) essentially a perturbation of the identity wedefine the following concept of anisotropy.

Definition 4. For each ρ ∈ D1(0) the anisotropy of the operator M τ1,τ2ρ is

defined by ǫτ1,τ2(ρ) := ||M τ1,τ2ρ − I||.

The following properties are immediate: ǫτ1,τ2(ρ) = ǫτ1,τ2(|ρ|), and 0 ≤ǫτ1,τ2(ρ) ≤ 2. Intuitively, ǫτ1,τ2(ρ) is an increasing function of |ρ|. By (7.9)the anisotropy of the operatorMρh(a) is given by ǫr2,r2(ρh(a)) = ǫr1,1(ρh(a)).

Let us first consider the family of sections σλ. For these sections weobtain the following parameters

δh,λ(t) =(1 + h)(1 − t2)(1− λ2)

(1 + λ2)(ht2 − 2t+ h) +√kh,λ(t)

, (7.10)

and

ρh,λ(t) = −2(1− h)(1 + t)λ(√kh,λ(t) + (1 + λ2)((1 + t2)h− 2t))√

1− h2(1− t)((1 + λ2)√kh,λ(t) + Ch,λ(t))

i, (7.11)

with

kh,λ(t) = (t2 − 2ht+ 1)2(1 + λ4) + 2((2h2 − 1)t4 − 4ht(1 + t2) + 6t2 − 1 + 2h2)λ2;

Ch,λ(t) = (t2 − 2ht+ 1)(1 + λ4) + 2((2h− 1)(1 + t2) + 2(h− 2)t)λ2.

For all h ∈] − 1, 1[ and λ ∈ [0, 1[, the parameter δh,λ behaves likea dilation parameter since 0 < δh,λ(t) < ∞, for all t ∈] − 1, 1[, withlim

t→−1+δh,λ(t) = 0 and lim

t→1−δh,λ(t) = ∞. With respect to the parameter

ρh,λ, it holds limt→−1+

ρh,λ(t) = 0 and limt→1−

ρh,λ(t) = 0, for all h ∈]− 1, 1[ and


λ ∈ [0, 1[. Thus, the function ϕρh,λ(t)(z) reduces to the identity function andMρh,λ(t) reduces to the identity operator, for small and large scales.

Lemma 9. For each u ∈ R+ the solution of the equation δh,λ(t) = u in

order to t is given by t = ph,λ(u), with

ph,λ(u) =−4u(1 + λ2) + 2

p

Dh,λ(u)

2(λ2Bh(u) + (1− h)u2 − 2hu(1 + λ2)− (1 + h)), (7.12)

where Dh,λ(u) := (1 + λ2)2((h− 1)u2 − (1 + h))2 − 4λ2(Bh(u))2, Bh(u) :=

(h − 1)u2 + 1 + h. Moreover, the function p is strictly increasing for allh ∈]− 1, 1[ and for all λ ∈ [0, 1[.

We define now the concept of global anisotropy of each section σλ,which characterizes in a sense how much each section σλ deviates from thefundamental section.

Definition 5. For each λ ∈ [0, 1[ the global anisotropy of the section σλis defined by Eh,λ :=

∫ 1−1 ǫ

r1,1(ρh,λ(t)) dt.

As for each h ∈]− 1, 1[ and λ ∈ [0, 1[ we have 0 ≤ ǫr1,1(ρh,λ(t)) ≤ 2, forall t ∈]− 1, 1[, then 0 ≤ Eh,λ ≤ 4. Thus, the concept of global anisotropy isa concept with an important meaning. We remark that it is also possible todefine this concept directly on the unit sphere.

Taking into account our considerations and the intertwining relation(7.3) we will modify the condition (5.1) and we will study the square in-tegrability of the system {RsRsh,σλDσλψ}, with ψ being a function withsupport on a fixed cap Uh. Our intention is to transfer the study of theadmissibility condition on S2 to the tangent plane and to relate wavelets onS2 with wavelets on its tangent plane.

Theorem 4. Let h ∈]− 1, 1[, λ ∈ [0, 1[ be fixed and ψ ∈ L2(S2) such that∅ 6= supp(ψ) ⊂ Uh with

0 6= cψ =

∫

R2

|Θψ(x)|2|x|2 dx <∞, (7.13)

where Θψ denotes the Fourier transform of Θψ. Then ψ is an admissiblefunction for the section (s, σλ).

28 Milton Ferreira

Proof. For an arbitrary f ∈ L2(S2) we have∫

Spin(3)

∫ 1

−1|⟨f,RsRsh,σλDσλψ

⟩L2(S2)

|2dµ(σλ(t))dµ(s)

=

∫

Spin(3)

∫ 1

−1|⟨Rsf,Rsh,σλDσλψ

⟩L2(S2)

|2dµ(σλ(t))dµ(s)

=

∫

Spin(3)

∫ 1

−1|⟨Θ(Rsf),Θ(Rsh,σλDσλψ)

⟩L2(R

2)|2dµ(σλ(t))dµ(s)

=

∫

Spin(3)

∫ 1

−1|⟨Θ(Rsf),Mρh,λ(t)D

δh,λ(t)Θψ⟩L2(R

2)|2dµ(σλ(t))dµ(s)

=

∫

Spin(3)

∫ 1

−1|⟨M−ρh,λ(t)Θ(Rsf),D


2)|2dµ(σλ(t))dµ(s).

(7.14)

For each s ∈ Spin(3) and t ∈]− 1, 1[ we have

|⟨M−ρh,λ(t)Θ(Rsf), D

δh,λ(t)Θψ⟩|2 ≤ sup

t′∈]−1,1[

|⟨M−ρh,λ(t′)Θ(Rsf), D

δh,λ(t)Θψ⟩|2

(7.15)and

inft′∈]−1,1[

|⟨M−ρh,λ(t′)Θ(Rsf), D

δh,λ(t)Θψ⟩|2 ≤ |

⟨M−ρh,λ(t)Θ(Rsf), D

δh,λ(t)Θψ⟩|2.

(7.16)

Then we obtain

inft′∈]−1,1[

∫

Spin(3)

∫ 1

−1|⟨M−ρh,λ(t′)Θ(Rsf),D

δh,λ(t)Θψ⟩|2dµ(σλ(t))dµ(s) ≤

∫

Spin(3)

∫ 1



supt′∈]−1,1[

∫

Spin(3)

∫ 1

−1|⟨M−ρh,λ(t′)Θ(Rsf),D

δh,λ(t)Θψ⟩|2dµ(σλ(t))dµ(s). (7.17)

Since the operators Θ, Rs, and Mρh,λ(t′) are unitary and bijective map-

pings, for each f ∈ L2(S2) we can find g ∈ L2(S2) such thatM−ρh,λ(t′)Θ(Rsf) =Θ(Rsg),

Let us first consider the integral∫

Spin(3)

∫ 1

−1|⟨Θ(Rsg),D


2)|2dµ(σλ(t))dµ(s). (7.18)

We claim that there exist constants 0 < C1 ≤ C2 <∞ such that

C1||g||2 ≤∫

Spin(3)

∫ 1

−1

|⟨Θ(Rsg), D


2)|2dµ(σλ(t))dµ(s) ≤ C2||g||2.

(7.19)


for all g ∈ L2(S2).

Considering the change of variable δh,λ(t) = u and using (7.12), themeasure dµ(σλ(t)) changes to dµ(u) = gh,λ(u)du, with

gh,λ(u) =2(1 +

λ(1−ph,λ(u)2)1+ph,λ(u)2λ2

− ph,λ(u)(1+λ2)

1+ph,λ(u)2λ2

)

(1 +

λ(1−ph,λ(u)2)1+ph,λ(u)2λ2

+ph,λ(u)(1+λ2)1+ph,λ(u)2λ2

)3 p′h,λ(u).

The new measure is equivalent to the measure du/u3 in the sense that thereare constants 0 < A1 ≤ A2 <∞ independent of our parameters such that

A11

u3≤ gh,λ(u) ≤ A2

1

u3, for all u ∈ R

+, h ∈]− 1, 1[, λ ∈ [0, 1[.

Therefore, (7.19) is equivalent to

C1A1||g||2 ≤∫

Spin(3)

∫ +∞

0

| 〈Θ(Rsg), DuΘψ〉

L2(R2)|2 duu3︸︷︷︸

I

dµ(s) ≤ A2C2||g||2.

(7.20)

For each s ∈ Spin(3) we extend the integral (I) considering

(b, θ) 7→∫ +∞

0|⟨T−bR−θΘ(Rsg),D

uΘψ⟩|2 duu3, (7.21)

where T−b and R−θ are the usual translation and rotation operators associ-ated to the CWT on R

2, defined respectively by

T bF (x) = F (x− b), b ∈ R2; RθF (x) = F (R−θx), R−θ ∈ SO(2).

The function (7.21) is continuous and integrable because Θψ fulfils condition(7.13). Using results of the 2D CWT we obtain

∫

R2

∫

SO(2)

∫ +∞

0

|⟨Θ(Rsg), T

bDuRθΘψ⟩|2 duu3dθdb = CΘψ ||Θ(Rsg)||22 = CΘψ ||g||22.

Hence, one has that

0 <

∫ +∞

0| 〈Θ(Rsg),D

uΘψ〉 |2 duu3

<∞ a.e., ∀s ∈ Spin(3), ∀ 0 6= g ∈ L2(S2)

and by the Dominated Convergence Theorem, the function

s 7→∫ ∞

0| 〈Θ(Rsg),D

uΘψ〉 |2 duu3

is continuous on the compact group Spin(3). This implies condition (7.20).

30 Milton Ferreira

Thus, returning to (7.17) there exist constants 0 < D1 ≤ D2 <∞ suchthat

D1 inft′∈]−1,1[

||M−ρh,λ(t′)Θ(Rsf)||2 ≤

≤∫

Spin(3)

∫ 1



≤ D2 supt′∈]−1,1[

||M−ρh,λ(t′)Θ(Rsf)||2

which implies that

D1||f ||2 ≤∫

Spin(3)

∫ 1

−1

|⟨M−ρh,λ(t)Θ(Rsf), D

δh,λ(t)Θψ⟩|2dµ(σλ(t))dµ(s) ≤ D2||f ||2.

This theorem shows that one can recover a correspondence principlebetween spherical wavelets with compact support on caps and Euclideanwavelets defined on discs.

It is possible to establish similar results for the sections σc. ApplyingLemma 6 we obtain the parameters

δh,c(φ) =(1 + h)(1 − |σc|2)

(1 + |σc|2)h+ 2cosφ+√kh,c(φ)

, (7.22)

and

ρh,c(φ) = −2c sin φ(1− h)((1 + |σc|2)h+ 2cos φ+√kh,c(φ))√

1− h2((1 + |σc|2 + 2cosφ)√kh,c(φ) + Ch,c(φ))

i, (7.23)

with

kh,c(φ) := 4(cosφ+ h)2c2 sin2 φ+ ((1− c2) sin2 φ+ 2 cosφ(cosφ+ h))2;

Ch,c(φ) := kh,c(φ) + 2(1− h)((1 + |σc|2 + 2h cosφ) cosφ+ 2hc2 sin2 φ).

For all h ∈]− 1, 1[ and c ∈ [0, 1[, the parameter δh,c behaves like a dila-tion parameter since 0 < δh,c(φ) <∞, for all φ ∈]0, π[, with lim

φ→0+δh,c(φ) = 0

and limφ→π−

δh,c(φ) = ∞. Moreover, limφ→0+

ρh,c(φ) = 0 and limφ→π−

ρh,c(φ) = 0.

Thus, the function ϕρh,c(φ)(z) reduces to the identity function and Mρh,c(φ)

reduces to the identity operator, for small and large scales.


The next lemma is easily achieved if we consider a change of variablesin the sections σc. Putting t = − cosφ, the sections σc(φ) can be describedas σc(t) = (0, c

√1− t2, t), t ∈]− 1, 1[.

Lemma 10. For each u ∈ R+ the solution of the equation δh,c(φ) = u in

order to φ is given by cosφ = ph,c(u), with

ph,c(u) =2u− [Ch(u)

2(1 + u)2c4 − 2Ch(u)Dh(u)(1 + u)c2 + ((h− 1)u2 − 1− h)2]1/2

Ch(u)(1 + u)(c2 − 1),

(7.24)

where Ch(u) := (h− 1)u+1+h and Dh(u) := (h− 1)u2 +h+1. Moreover,the function p is strictly decreasing for all h ∈]− 1, 1[ and for all c ∈ [0, 1[.

Theorem 4 can be also formulated to sections σc if the quasi-invariantmeasure is chosen conveniently. There are many other sections that canbe studied and for which Theorem 4 holds. In a forthcoming paper wewill characterize all admissible global and local sections that can give riseto spherical continuous wavelet transforms. Moreover, the results of thissection can be easily generalized to the sphere Sn−1, for n > 3.

Acknowledgments

The author would like to thank Prof. Hans Feichtinger for his hospitalityand support during a visit at NuHAG, Department of Mathematics, Univer-sity of Vienna as well as Prof. J. P. Antoine and P. Vandergheynst for thesupport and discussions during a stay at the Institut de Physique Theoriqueof Universite Catholique de Louvain. The author is also very thankful to H.Rauhut, M. Fornasier, U. Kahler, P. Cerejeiras and N. Faustino for fruitfuldiscussions. The work was supported by PhD-grant no. SFRH/BD/12744/2003of Fundacao para a Ciencia e a Tecnologia of Portugal.

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Departamento de Matematica, Universidade de Aveiro,Campus Universitario de Santiago

3810-193 Aveiro, Portugal.E-mail: [email protected]

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