article.aascit.orgarticle.aascit.org/file/pdf/9210811.pdf · international journal of mathematical...

International Journal of Mathematical Analysis and Applications

2018; 5(3): 66-84

http://www.aascit.org/journal/ijmaa

ISSN: 2375-3927

Applications on Triagular Subgroups of Sp with Reproducing Groups

Simon Joseph1, *

, Manal Juma2, Isra Mukhtar

3

1Department of Mathematics, University of Juba, Juda, South Sudan 2Department of Mathematics, Prince Sattam bin Abdulaziz University, Alkharj, Kingdom of Saudi Arabia 3Department of Mathematics, Shaqra University, Al-Riyadh, Kingdom of Saudi Arabia

Email address

*Corresponding author

Citation Simon Joseph, Manal Juma, Isra Mukhtar. Applications on Triagular Subgroups of Sp with Reproducing Groups. International Journal of

Mathematical Analysis and Applications. Vol. 5, No. 3, 2018, pp. 66-84.

Received: June 28, 2018; Accepted: July 10, 2018; Published: September 3, 2018

Abstract: Consider the (extended) metaplectic representation of the semidirect product 1 , , between the Heisenberg group and the symplectic group. Subgroups Σ D, with Σ being a 1 1

symmetric matrix and D a close subgroup of 1 , , are the main concerned. They shall give a general setting for the

reproducibility of such groups which include and assemble the ones for the single examples treated in Cordero et al.(2006) [3].

As a byproduct, the extended metaplectic representation restricted to some classes of such subgroups is either the Schrodinger

representation of or the wavelet representation of D with closed subgroup of 1 , by E. Cordero,

A. Tabacco [11]. Finally, we shall provided new examples of reproducing groups of the type Σ D in dimension = 1.

Keywords: Reproducing Formula, Metaplectic Representation, Wigner Distribution, Semidirect Product

1. Introduction

Reproducing formulae appear almost everywhere in the

literature, from coherent states in physics to group

representations, Gabor analysis, wavelet analysis and its

many generalizations. This theory has a wealth of

applications in engineering, physics and numerical analysis

(see, e.g., [1, 2, 6, 7, 10] and references therein). It is

remarkable to observe that most existing reproducing

formulae for the sequence of functions can

be formulated in one and the same general form, that are, an

integral of the series formula of the types

∑ ∑ , !"#$ !

% "#&", '( )** +, (1)

in the following sense. First of all, the domain of integration

H is a (connected, closed, Lie) subgroup of the semidirect

product = Sp(1 , ) between the Heisenberg

group and the symplectic group Sp(1 , ), dh is a

left Haar measure of H, and ! are the extended metaplectic

representation of , to be defined below in detail. The

sequence of functions # ∈ () are usually referred

as wavelet or admissible vector. As is showed below by E.

Cordero, A. Tabacco [11], many standard reproducing

formulae, such as those arising in Gabor analysis and wavelet

theory, are of not is both relevant and difficult, and is the

main theme of our recent investigations [3, 4], together with

the above type: for some groups H, but not for all, one finds

# ∈ ) in such a way that the above formula holds

weakly for every ∈ (). Thus, some groups H give

rise to interesting analysis and some do not, like, for

example, the full factors themselves, i.e. - or

Sp(1 , ) - . The question whether a given subgroup

H - leads to a reproducing formula or the explicit

description of the admissible vectors. The ongoing

formulation of things maybe simplified a bit because the

reproducing formula (1) is equivalent to

∑ //

∑ % &", '( )** +, (2)

which is manifestly insensitive to phase multiplicative

factors, that is, invariant under transformations of the

International Journal of Mathematical Analysis and Applications 2018; 5(3): 66-84 67

types # ⟼ 12(3)#. This allows a technical reduction, that

is, one can factor out the center of the Heisenberg group,

whose action through ! are through phase factors, and one

may safely pass from the whole group to the somewhat

simpler group = ℍ() ⋊ Sp(1 + , ℝ). The formalize

this discussion in the following:

Definition 1. A connected Lie subgroup H of G = ℝ() ⋊ Sp(1 + , ℝ) is a reproducing group for ! if there

exists the sequence of functions # ∈ (ℝ()) such that

the identity (1) holds weakly for all ∈ (ℝ()). All

such # ∈ (ℝ() ) are called reproducing sequence of

functions.

Notice that we do require formula (1) to hold for all

functions in (ℝ()) for the same of sequence of windows # , but we do not require the restrictions of ! to H to be

irreducible.

Consider a class of triangular subgroups of the symplectic

group, that collectively denote as the class ℰ . From the

structural point of view, a group H ∈ ℰ is a semidirect

product of the form

H = Σ ⋊ D, where Σ is a (1 + )-dimensional subspace

of (1 + ) × (1 + ) symmetric matrices and D is a closed

subgroup of GL(1 + , ℝ) acting on Σ; hence Σ are abelian

and normal in H. Evidently, when seen within G, each of

these groups is contained in the symplectic factor. We shall

show that many examples used in the applications fall in this

class. This motives the study of the class ℰ. The main result is provided by Theorem 13 below, which

contains necessary and sufficient conditions for the sequence

of wavelets 5 to be reproducing on a group H in ℰ [11]. This

result is a far-reaching extension of the reproducing

conditions for the special cases treated in [4]. Underline that

this result is based on a deep study of the properties of a

quadratic mapping Φ on ℝ() , we shall provide new

examples of reproducing groups in the class Ԑ in dimension

= 1.

We contain some additional remarks concerning an

alternative formulation of the concept of admissible group, a

formally stronger notion than the notion given in Definition

1. Also clarify in what sense the setup includes Gabor and

wavelet analyses (see, e.g., [11]).

2. Preliminaries and Notation

The symplectic group is defined by

Sp(1 + , ℝ) = g∈ GL(2(1 + ), ℝ): g() (J) g = J,

where

8 = 9 0 ;()−1() 0 =

is the standard symplectic form

> (∑ ?@@ , y) = ∑ ?@ @()(J) B , ∑ ?@@ , B ∈ ℝ() (3)

The metaplectic representations of (the two-sheeted

cover of) the symplectic group arises as intertwining operator

between the standard Schrödinger representation ρ of the

Heisenberg group ℍ() and the representation that is

obtained from it by composing ρ with the action of

Sp(1 + , ℝ ) by automorphisms on ℍ() . We briefly

review its construction.

The Heisenberg group ℍ() is the group obtained by

defining on ℝC the products

DE , 1 + ). (Ej)F , G1 + HI = DEj + Ej, D + 2 + 1 I + 12 >(Ej, (Ej)F )I

where > stands for the standard symplectic form in ℝ() given in (3). Denote the translation and modulation operators on (ℝ()) by

L(∑ MNO )N (1 + ) = ((1 + )−∑ ?@@ ) and P(∑ QNR )N (1 + ) = 1S2∑ QNRN ,$(1 + )

The Schrödinger representation of the group ℍ() on (ℝ()) is then defined By

T UV ?@@ , V ξ@@ , 1 + X (B) = 1S2()1YS2∑ MNRN ,∑ QNRN $1S2∑ QNRN ,ZR$ UB − V ?@@ X= 1S2()1S2∑ MNRN ,∑ QNRN $L(∑ MNO )N P(∑ QNRN )(B)

where write E = ( ∑ x@@ , ∑ ξ@@ ) when we separate space

components (that are ∑ x@@ ) from frequency components

(that are ∑ ξ@@ ) in points Ej in phase space ℝ() . The

symplectic group acts on ℍ() via automorphisms that

leave the center (0, 1 + ): (1 + ) ∈ ℝ ∈ ℍ() ≃ ℝ of ℍ() pointwise fixed:

· (Ej, 1 + ) = ( Ej, 1 + ).

Therefore, for all fixed ∈ Sp( 1 + , ℝ ) there are

representations

T_R: ℍ() → a ((ℝ())), (Ej, 1 + ) ⟼ T+ · (Ej, 1 + ),

Whose restriction to the center is a multiple of the identity.

By the Stone–von Neumann theorem, T_R are equivalents to T . That is, there exists an

intertwining unitary operator

68 Simon Joseph et al.: Applications on Triagular Subgroups of Sp with Reproducing Groups

( ) ∈ a((ℝ())) such that T_R(Ej, 1 + ) = ( ) ∘ T(Ej , 1 + ) ∘ ( )−1

, for all (Ej , 1 + ) ∈ ℍ() . By

Schur’s lemma, are determined up to a phase factors 12dR , e∈ ℝ. It turns out that the phase ambiguity is really a

sign, so that lifts to a representation of the (double cover

of the) symplectic group. It is the famous metaplectic or

Shale–Weil representation.

The representations T and can be combined and give

rise to the extended metaplectic representation of the group G

= ℍ() ⋊ Sp( 1 + , ℝ ), the semidirect product of ℍ() and

Sp(1 + , ℝ). The group law on G are

f(E , 1 + ), g ∙ DGEjF , f gH , ^j

F I = D+Ej, 1 + ,. G EjF , f gH , ^j

F I (4)

and the extended metaplectic representations ! of G are

!+(Ej, 1 + ), , = T(Ej, 1 + ) ∘ ( ). (5)

Observe that the reproducing sequence of formula (1) is insensitive to phase factors: if we replace !(h) # with 12dR !(h)#

the formula is unchanged, for all e ∈ ℝ. The role of the center of the Heisenberg group is thus irrelevant, so that the “true”

group under consideration is ℝ() ⋊ Sp(1 + , ℝ), whichwe denote again by G. Thus G acts naturally by affine transformations on phase space,

namely

g· (∑ x@@ ,∑ ξ@@ ) = +(1 + , 1 + ), ,· (∑ x@@ ,∑ ξ@@ )= (∑ x@@ , ∑ ξ@@ ) () + (1 + , 1 + ). () (6)

For elements Sp(1 + , ℝ) in special form, the metaplectic representation can be computed explicitly in a simple way. For ∈ (ℝ()) , we have

Di 00 ^() YjI +∑ ?@k@ , = (&1l )Y ⁄ + Y ∑ ?@k@ , (7)

f n1 0o 1pg +∑ ?@k@ , = ±1Y2Sr ∑ sNRN ,∑ sNRN $+∑ ?@k@ , (8)

(8) = t() ⁄ ℱY (9)

where ℱ denotes the Fourier transforms

ℱ V UV ξ@@ X = v V UV ?@k@ X 1YS2∑ sNRN ,∑ QNRN $ℝ(wxy)

&(V ξ@)@ , ∈ (ℝ()) ∩ (ℝ())

In the above formula and elsewhere, ∑ x@@ , ∑ ξ@@ $ denotes

the inner products of ∑ ?@k@ , ∑ ξ@@ ∈ ℝ(). Similarly, for , g∈ +ℝ(),, , g $ will denote their inner product in (ℝ()). Other notation is as follows. Put ℝ = ℝ \ 0, ℝ±

= (0, ±∞). For 0 ≤ ≤ ∞,∙ stands for the -norm of measurable functions on ℝ() with respect

to Lebesgue measure. The left Haar measure of a group H

will be written dh and we always assume that the Haar

measure of a compact group is normalized so that the total

mass is one. Let Ω be an open set of ℝ(). Then o~ (Ω) is

the space of smooth functions with compact support

contained in Ω.

3. Gabor and Wavelet Analyses

We show below that both Gabor and wavelet analyses can

be viewed as particular cases of the general setup that we are

considering. This fact is somehow know and is perhaps part

of common knowledge, but on the one hand we could not

locate it in the literature in a precise fashion and, on the other

hand, we want to present some additional remarks that are of

some independent interest. Thus start with a side observation

(see, e.g.., [11]).

3.1. Weak Admissibility

The notion of reproducing group admits an equivalent

version that is obtained by weakening a property that has

been investigated in [3] and that is formulated by means of

the Wigner distribution. The cross-Wigner distributions R,R of , g ∈ (ℝ()) are given by

∑ R,R +∑ ?@k@ , ∑ ξ@@ , = ∑ 1YS2∑ QNRN ,R$ f∑ ?@k@ + R g gj f∑ ?@j@ − j g &y (10)


The quadratic expressions R :=R,R are called the Wigner distributions of . Collect below some of its well-known

properties.

Proposition 2. The Wigner distribution of ∈ (ℝ()) satisfies:

(i). R are uniformly continuous on ℝ(), and ∑ R ≤ 2()/∑ /.

(ii). R are real-valued.

(iii). Moyal’s identity: R , R $(ℝ(wxy)) = , g$+ℝ(wxy),j, gj$(ℝ(wxy)) (iv). If ∈ (ℝ()), then R∈ (ℝ()). (v). Marginal properties:

∑ R +∑ ?@k@ , ∑ ξ@@ , & (∑ ξ@@ℝ(wxy) ) = ∑ (∑ ?@j@ ) , ∑ R +∑ ?@j@ , ∑ ξ@@ , &(∑ ?@j )@ = ∑ j

(∑ ξ@@ )ℝ(wxy) (11)

for j ∈ (ℝ()), ∈ (ℝ()), respectively.

(vi). If both ,j are in (ℝ()) (hence in (ℝ())) then

∑ R (>)&>ℝ(wxy) = ∑ // (12)

In [3] it is introduced the notion of admissible group, one for which there exists # ∈ (ℝ()) such that (15) below holds.

Together with some additional integrability and boundedness property of ℎ ⟼ R (ℎY ∙ +∑ x@@ , ∑ ξ@@ ,), it implies that a

subgroup H of G = ℝ() ⋊ Sp((1 + ), ℝ) is reproducing.

For the reader’s convenience, recall the statement of [3, Theorem 1], where the main point is made.

Theorem 3. Suppose that # ∈ (ℝ()) are such that the mapping

ℎ ⟼ R()R +∑ ?@j@ , ∑ ξ@@ , = R fℎY. +∑ ?@j@ , ∑ ξ@@ ,g (13)

is in (H) for a.e. +∑ ?@j@ , ∑ ξ@@ , ∈ ℝ() and

∑ R fℎY. +∑ ?@j@ , ∑ ξ@@ ,g% &ℎ ≤ P, '( ). 1. +∑ ?@j@ , ∑ ξ@@ , ∈ ℝ() (14)

Then condition (1) holds for all ∈ (ℝ()) if and only if the following admissibility condition are satisfied:

∑ R fℎY. +∑ ?@j@ , ∑ ξ@@ ,g% &ℎ = 1, '( ). 1. +∑ ?@j@ , ∑ ξ@@ , ∈ ℝ() (15)

Assumptions (13) and (14) are sufficient but not necessary conditions for a subgroup to be reproducing, see the examples

related to Gabor and wavelet analysis in the next subsections.

The notion of weak admissible group is as follows. Consider first the vector space

= e) R , ∈ (ℝ()) = ∑ (R) , () ∈ +ℝ(),, ∈ ℂ (16)

For , g ∈ +ℝ(), , a straightforward computation

gives

R,R = 9RR + tR2R − (1 + t)(R + R )= (17)

Since span R,R, , g ∈ +ℝ(), is dense in +ℝ(),(see [3]), it follows from (17) that also V is dense

in +ℝ(),.

Assume that the subgroup H is reproducing and let # ∈(ℝ()) be reproducing sequence of functions. Define the

conjugate-linear functionals ℓ on V by

∑ ℓ() = ∑ R()R , $ &ℎ% , ∀ ∈ (18)

The functionals ℓ are well-defined and continuous on V

with respect to the norm, as shown presently. Let = ∑ (R), () ∈ +ℝ(),, ∈ ℂ, be an element of

the space V. Using Moyal’s identity (Proposition 2) and the

reproducing condition (2), we have


Vℓ() = V v R()R , V (R)

$ &ℎ%

= V V v R()R , (R) $ &ℎ

%

= V V

v +, , !(ℎ)#$%

&ℎ = V V /()/

= V V

v (R) UV ?@j@ , V ξ@@ X &(V ?@j@ )&(V ξ@)@ℝ(wxy)

= V v V (R) UV x@@ , V ξ@@ X &(V ?@j@ )&(V ξ@)@ℝ(wxy)

= V 1, V (R) $

= v V j¡ (V ?@j@ , V ξ@@ )&(V ?@j@ )&(V ξ@)@ℝ(wxy)

≤ ¢V ¢

Moreover, from the previous computations it is clear that ℓ() = 1, $, for every ∈ V. Finally, since V is dense

in (ℝ()), the functional ℓcan be uniquely extended to

a continuous functional (ℓj) £ on (ℝ()), which coincides

with 1 ∈ (ℝ())

Observe that, in general, this does not imply that ∑ (ℓj) £ () = % ∑ R()R , $ dh,

∀ ∈ (ℝ()). Indeed, one should know that the mapping

→R()R , $ is in (H) in order for the integral to make

sense, and it should satisfy

v V R()R &ℎ, $ = 1, $, ∀ ∈ +ℝ(),.%

Those observations yield to the following definition (see,

g.e., [11]).

Definition 4. Say that the subgroup H of G is weakly

admissible if there exists the sequence of functions # ∈(ℝ()) such that the functional (18) is well-defined on the

set V defined in (16) and verify

∑ ℓ() = ∑ R()R , $ &ℎ = 1, $, ∀ ∈ % (19)

In a position to state and prove the first observation.

Theorem 5. The following are equivalent:

(i) H is weakly admissible and (19) holds for # ∈(ℝ()); (ii) H is reproducing and # ∈ (ℝ()) are reproducing

sequence of functions.

Proof. Observe that the reproducibility condition (2) can

be checked on the dense subspace

(ℝ()) ⊂ (ℝ()). (i) ⇒ (ii). Take ∈ (ℝ()) and use the admissibility

condition with = (R). Then, Moyal’s identity and ;, (R) $ = ∑ // immediately provide the desired result.

(ii) ⇒ (i). It follows by the previous observations. ∎

Remark. If the assumptions (13) and (14) of Theorem 3

hold, then the mappings → R()R , $ is in () (one can apply Fubini’s

theorem and exchange the integrals) and, moreover, ∑ R()R , $ &ℎ = ∑ R()R

&ℎ, % $ =%1, $, ∀ ∈ (ℝ()). Hence the weak admissibility

condition generalize (15)

3.2. Gabor Analysis

Gabor’s reproducing of the series formula are given by

V = v V , L(∑ MNj )N P(∑ QNR )N 5$ L(∑ MNj )N P(∑ QNRN )5 &(V ?@j )@ &(V ξ@@ℝ(wxy))

which are (weakly) true for every 5 ∈ ( ℝ() ),

with ∑ /5/ = 1. We shall refer to this basic fact as to

Gabor’s theorem. The extended metaplectic

representations μ! restricted to the subgroup

H ≅ ℝ() consisting of the first factor in G is, of course,

the Schrödinger representation. If consider the sequence of

functions 5 ∈ (ℝ()) ∩ ℱ(ℝ()), so that conditions

(13) and (14) are satisfied, the admissibility condition (15)


becomes ∑ R f∑ x@@ – (1 + ), ∑ ξ@@ – (1 + )g d(1 +ℝ(wxy))d(1 + ) = 1, for a. e. (∑ x@@ , ∑ ξ@@ ) ∈ ℝ(), that are

∑ R (1 + , 1 + )&(1 + )&(1 + ) = 1ℝ(wxy) (20)

The need the conditions (13) and (14) to have R ∈ ( ℝ() ) for granted. However, if The drop the

requirements R ∈ ( ℝ() ), the reproducing the

sequence of windows 5 can be rougher, as shown below.

Proposition 6. Let 5 ∈ (ℝ()) with ∑ /5/ =1

(i) If 5 ∈ ( ℝ() ), then ℝ(wxy)(ℝ(wxy) ∑ R (1+ ,

1+)d(1+))d(1+) =1

(ii) If 5j ∈ (ℝ() ), then ℝ(wxy)(ℝ(wxy) ∑ R (1+ ,

1+)d(1+))d(1+) =1

Proof. If 5 ∈ (ℝ() ) with ∑ /5/ = 1, then 5 are

reproducing by Gabor’s theorem. By the second marginal

property in (11), the map (1+ ) → R(1+ , 1+ ) is

integrable on ℝ() for a.e.

(1+) ∈ ℝ(). Since

V5j

(1 + ) = v V R

(1 + , 1 + )d(1 + )ℝ(wxy)

and

v V5j

(1 + ) d(1 + ε)ℝ(wxy) = V/5

j / = V/5/ = 1,

the claim is proved. If 5j ∈ ( ℝ() ), use the same

arguments with the first marginal property in (11). ∎

Remark. The previous proposition indicates that (20) may

fail to be true even for reproducing sequence of

functions 5 ∈ +ℝ(), , but it is replaced by subtly

weaker conditions for integrable reproducing sequence of

functions (or for reproducing sequence offunctions with

integrable Fourier transform). Assumptions (13) and (14) are

actually not necessary for 5 to be reproducing sequence of

functions, as the simple example below illustrates.

Example 7. In dimension = 0, consider the box sequence

of functions 5 (∑ ?@j@ ) = χ[−1/2,1/2](∑ ?@j@ ). Then 5 ∈ (ℝ)

and ∑ /5/ = 1, so that it is a reproducing sequence of

function. On the other hand, conditions (13) and (14) are not

fulfilled. This is seen by computing the Wigner distributions R. Indeed, using the definition (10), we get

R UV ?@j@ , V ξ@@ X =

±²²²²²³²²²²²

etµ2¶+1 + 2 ∑ ?@j@ , ∑ ξ@@ ·¶ ∑ ξ@@ , V ?@j@ ∈ (−12 , 0), V ξ@@ ≠ 0 2 U1 + 2 V ?@j@ X , V ?@j@ ∈ (−12 , 0), V ξ@@ = 0

etµ2¶+1 − 2 ∑ ?@j@ , ∑ ξ@@ ·¶ ∑ ξ@@ , V ?@j@ ∈ [0, 12), V ξ@@ ≠ 0 2 U1 − 2 V ?@j@ X , V ?@j@ ∈ [0, 12), V ξ@@ = 0

0, 'lℎ1(ºte1

Clearly, R ∉ (ℝ2), however, observe that Proposition

6 is satisfied with the admissibility conditions ℝ(ℝ ∑ R (∑ ?@j@ , ∑ ξ@@ ) d(∑ ξ@@ ) d(∑ ?@j )@ = 1.

The Gabor case is a particular example of a subgroup of

the form H = ℝ() ⋊ K, with K subgroup of Sp(1 + ,ℝ)

(here K = ;()). The reproducibility of H is equivalent to

asking the compactness of K. Indeed, if arbitrary translations

are allowed in the affine action of H, then the symplectic

factor must be compact, as shown below (see, e.g., [11]).

Proposition 8. If H = ℝ() ⋊ K, with K ⊂ Sp(1+, ℝ),

then H is reproducing if and only if K iscompact.

Proof. The left Haar measure of H is given by d ℎ =

d (∑ ?@j )@ d (∑ ξ@@ ) dk, where d (∑ ?@k )@ , d (∑ ξ@@ ) are the

Lebesgue measure on ℝ() and dk is the left Haar measure

of K. In the computations below, takes ∈ (ℝ()). First

write the right-hand side of (2), then apply Plancherel’s

theorem, compute the Fourier transform of the time-shifts L(∑ sNR )N , then use the Parseval identity and, finally, the

Fourier transform of the frequency-shift P(∑ QNRN ):


v v V , L(∑ MNj )N P(∑ QNR )N (¼)#$ &(V ?@j )@ &

ℝ(wxy)½(V ξ@)@ dk

= v v V j

, ℱ fL(∑ MNj )N P(∑ QNRN ) (¼)#g$ℝ(wxy)½

&(V ?@j )@ &(V ξ@)@ dk =

v v V v j

(1 + )1S2f∑ MNjN g()ℱ(P(∑ QNj )N j(¼)#j

ℝ(wxy)

(1 + )&(+ )

&(V ?@j )@ &(V ξ@j )@ dk ℝ(wxy)½

= v v v V ℱY(j

ℱ(P(∑ QNj )N j(¼)#j) UV ?@j@ X

ℝ(wxy)ℝ(wxy)½&(V ?@j )@ &(V ξ@j )@ dk

= v v v Vj

(¿)ℝ(wxy)ℝ(wxy)½

ℱ(P(∑ QNj )N (¼)#)(¿) &¿ &(V ξ@j )@ dk

= v À v À v V ℱ U (¼)#)(¿ − V ξ@j@ )X ℝ(wxy)

&(V ξ@j@ )Á j

(¿) &¿ℝ(wxy)

Á½

&¼

= À v j

(¿)&¿ℝ(wxy)

Á v V/ℱ+ (¼)#),/&¼

= V// v/+ (¼)#),/ &¼

= V///#/ v &¼ =

V// /#/ Âte(Ã)

The interchange in the order of integration is justified by Fubini’s theorem, since

v v Vj

(¿) ℱ(P(∑ QNj )N (¼)#(¿) &(V ξ@j )@ &¿ ≤ V/#/ℝ(wxy)ℝ(wxy)

//

The Haar measure of the locally compact subgroup K is finite if and only if K is compact. This concludes the proof. ∎

3.3. Wavelet Analysis

Examine the (generalized) continuous wavelet transform in higher dimensions, that is in ℝ(), with > 0. It arises from

restriction of the metaplectic representation to semidirect products of the form ℝ() ⋊ D, where D is any closed subgroup of

GL(1 + , ℝ). The product law is

(1+, a).( , )) = DfÅ() g , a ) I, 1+, ∈ ℝ(), a, ) ∈ D. (21)

When D = GL(1 + , ℝ) it is the so-called the Affine Group of Motions on ℝ() [9], and corresponds to the action (1 + ) ⟼ a(1 + ) + (1+). The wavelet representation of ℝ() ⋊ D on (ℝ()) associated with this action are given by

(ν (1+, a)5)(1 + ) = |det )|Y ⁄ 5 G)Y f1 + – (1 + )gH= (L()Å5)(1 + ), (1 + ) ∈ ℝ() (22)

the sequence of functions 5 ∈ (ℝ()) are admissible if the Calderón conditions are fulfilled:


∑ 5j

( )(∑ ξ@@() )) &) = 1, '( ). 1. ∑ ξ@@ ∈ ℝ()É (23)

where da is a left Haar measure on D. The subgroup D ⊂ GL(1 + , ℝ) may be identified with the subgroup of Sp(1 + , ℝ)

given by

Ê9) 00 )Y() = , ) ∈ Ë

And the metaplectic representations of D are

( (a)) (∑ ?@j@ ) = (det a)−1/2 (a −1(∑ ?@j )@ ) = ±|&1l)|Y ⁄ (a −1 (∑ ?@j@ )), ∈ (ℝ())

The group ℝ() ⋊ D is isomorphic to the subgroup H of ℝ() ⋊ Sp(1 + , ℝ) given by

= Êℎ(1 + , )) n1 + 0 p , 9) 00 )Y() = , (1 + ) ∈ ℝ(), ) ∈ Ë (24)

Observe that the group law with in Sp(d, ℝ) is ℎ(1+, a)h( , )) =hDfÅ() g , a ) I,

in accordance with (21). The extended metaplectic representation restricted H is

f !+ℎ(1 + , a),g(1 + ) = +ρ(1 + , 0) (a),(1 + ) = +L() (a),(1 + )

= ±|&1l)|Y ⁄ ()Y (1 + ) − (1 + ε)))= ±+L()Å,(1 + ). (25)

Thus, up to a sign, the extended metaplectic representation of H coincides with the wavelet representations ν, so that they

give rise to the same reproducing series formulas

∑ = ∑ , L()Å5$ℝ(wxy)É L()Å5 &ℎ(1 + , )) (26)

where dh are left Haar measure on H. If da is a left Haar measure of the group D and d(1+) is the standard Lebesgue measures

on ℝ(), a left Haarmeasure dℎ(1+, a) are given by

dh(1+, a) =d(1+)|det a|Y da.

One would expect that also the admissibility conditions related to the two representations coincide. The next result shows

the direct correspondence between them (see, e.g., [11]).

Proposition 9. Let 5 ∈ (ℝ()) ∩ (ℝ()). Then,

v V R%

UℎY. (V ?@j@ , V ξ@j@ )X &ℎ = v V 5j

( )(V ξ@j@() ))

&)É

In particular, the wavelet admissibility condition (23) and the Wigner one (15) coincide.

Proof. If 5 ∈ (ℝ()) ∩ (ℝ()), we have R ∈ C(ℝ()) and R(·,∑ ξ@j@ ) ∈ (ℝ()), for every ∑ ξ@@ ∈ ℝ(),

and the Wigner marginal property (11) holds. For h(1+, a)−1

= h(−a −1

(1+), a −1

) the action of H on the phase spaces are given

by

ℎ(1 + , ))Y ∙ UV ?@j@ , V ξ@j@ X = i)Y 00 )() jÍÎÎÎÏV ?@j@V ξ@j@ ÐÑÑ

ÑÒ + 9 −)Y(1 + )0 = =ÍÎÎÎÏ)Y(V ?@j@ − (1 + ))

)(V ξ@j )@() ÐÑÑ

ÑÒ

The change of variables a −1

(∑ ?@j@ – (1+)) = u, d(1+) = |det a | du, yields

v V R%

ÀℎY. UV ?@j@ , V ξ@j@ XÁ &ℎ = v À v V R À)Y ÀV x@j@ − (1 + )Á , )(V ξ@j@

() )Á &(1 + )ℝ(wxy)

ÁÉ

&)|det )|


= v v V R (Ó, )(V ξ@j@

() ))ℝ(wxy)É

&Ó &) = v V 5j ( )(V ξ@j@

() )) &)

É

that is the claim. ∎

Alike the Gabor case, the assumptions (13) and (14) are not necessary for a reproducing sequence of function. Indeed, it is

enough that R (·, ∑ ξ@j@ ) ∈ (ℝ()), for almost every

∑ ξ@j@ ∈ ℝ(). It is worthwhile observing that the wavelet reproducing formula associated to a given subgroup D can be obtained from

another subgroup of G, namely

Ô = ÊℎÕ(1 + , )) = G9 0−(1 + )= , 9 )Y() 00 )=H , (1 + ) ∈ ℝ(), ) ∈ Ë (27)

Indeed, g= (0, J) ∈ G, then g ℎ(1+, a)(g)Y = ℎÕ (1+, a) and H in (24) and Ô is conjugate. This implies that one is

reproducing if and only if the other is, and the corresponding reproducing formulae are equivalent. Observe that the extended

metaplectic representation of Ô is nothing else but the wavelet representation on the frequency side:

+( ! (ℎÕ (1 + , a) ),(1 + ) = +ρ(0, −(1 + )) ( )Y() ),(1 + )

= +PY() ( )() Y) , (1 + )

= ± GPY() Å(wxy) ÖwH(1 + ) = ±ℱ+L()Å ,(1 + ). (28)

4. The Class ×

Introduce a class of (lower) triangular subgroups of

Sp(1 + , ℝ) and derive general conditions for reproducing

formulae to hold true. From the structural point of view, a

group H ∈ ℰ is of the form H = Σ ⋊ D, where again D is a

closed subgroup of GL(1 + ,ℝ) that acts by automorphism

on ℝ() . Thus, given a homomorphism θ: D →

Aut( ℝ() ), that is, a (1 + ) -dimensional real

representation of D, and define the semidirect products

h(1+, a)ℎ f , )g = ℎ DfØ(Å)() g , a ) I (29)

The connection with Sp(1 + , ℝ) is as follows. First of

all, we shall identify D with Ê9) 00 )Y() = : ) ∈ Ë ⊂ (1 + , ℝ)

Secondly, preliminarily observe that

Ù = n 1 0Ú 1p : Ú ∈ BÂ(1 + , ℝ) ⊂ (1 + , ℝ) (30)

is an abelian Lie subgroup of Sp(1 + , ℝ), whereby the

matrix product amounts to Ú + Ú jF , the sum within the vector

space of (1 + ) by (1 + ) symmetric matrices, denoted by

Sym(1 + , ℝ). Then assume that we are given an injective

homomorphism of abelian groups

j:ℝ() → Sym(1 + , ℝ), (1+) ⟼ j (1+):= (Ú)

which establishes a group isomorphism of ℝ() with the

image Σ = j (ℝ() ) ⊂ Sym( 1 + , ℝ ). In other words,

assume that we are given a parametrization of a (1 + )-

dimensional subspace Σ of Sym(1 + , ℝ). Explicitly:

Σ = Ê9 1 0(Ú) 1= , (1 + ) ∈ ℝ()Ë ⊂ (1 + , ℝ) (31)

Finally, consider the products h = h(1+, a) defined by

ℎ(1 + , )) = 9 1 0(Ú) 1= 9) 00 )Y() = = 9 ) 0(Ú)) )Y() = (32)

Since

ℎ(1 + , ))ℎ DG1 + H , )I = 9 ) 0(Ú)) )Y() = i ) 0(Ú)f g) ( ()))Y() j

= i )) 0(Ú) ) ) + )Y() (Ú)f g) )Y () ( ))Y() j


= Û )) 0G(Ú) + )Y(Ú)f g)Y() H ) ) ()))() YÜ

and

ℎ GÝ()) G1 + H + (1 + ), ) )H = i )) 0(Ú)Ø(Å)f g (Å Å)(wxy) () ))Y() j

the semidirect product law (29) holds true if and only if the parametrization j and the representation

θ satisfy

(Ú)Ø(Å)()= )Y() (Ú))Y, a ∈ D,(1+ ) ∈ ℝ() (33)

Formally, a group ℰ can thus be described by a triple (D, j, θ), where D is a closed subgroup of

GL(1 + , ℝ), θ: D → Aut+ℝ(), is a representation, and j: ℝ() → Sym(1 + , ℝ), is injective homomorphism of

abelian groups; the data must satisfy the compatibility equation (33). Avoid this excess of notation and write directly H = Σ ⋊ D. If H ∈ ℰ is chosen, and hence D, assume that a left Haar measure da on D has been fixed and consequently the left Haar

measure on H that will be fixed is

&ℎ = & (1 + ) & )|&1l Ý())|, where & (1 + ) is the Lebesgue measure on ℝ(). Finally, the metaplectic representations restricted to H ∈ ℰ is given by

Àℎ(1 + , )) UV x@@ XÁ = G9 1 0(Ú) 1=H G9) 00 )Y() =H UV x@@ X

= ±1S2(ÞR)wxy ∑ sNRN ,∑ sNRN $ G9) 00 )Y() =H +∑ x@@ , = ±1S2(ÞR)wxy ∑ sNRN ,∑ sNRN $(&1l ))Y ⁄ +)Y ∑ x@@ , (34)

4.1. Examples

1. Let = 1. The Translation–Dilation–Sheering group (TDS) is the 4-dimensional triangular reproducing group introduced

and studied in [3]. It is defined by

L = ßÛ (1 + )ℓR 0(1 + )(à)ZRℓR f g ( ℓRY)() Ü : > −1, ℓ ∈ ℝ, B ∈ ℝá (35)

where if B = (B,B) ∈ ℝ and ℓ ∈ ℝ,

(à)ZR = i 0 BB Bj , ℓR = 91 ℓ0 1 =

It is isomorphic to the semidirect product Σ ⋊ D, with Σ= (Ú)ZR , B ∈ ℝ

D = (1 + ) ℓR , > −1, ℓ ∈ ℝ.

A simple computation shows that

θ ((1 + ) ℓR)B = (() (1 + ) ℓR)YB.

Thus TDS ∈ Ԑ. The group TDS is important because it is the group underlying shearlet theory (see e.g. [5, 8]). This

terminology stems from the geometric actions of ℓR on ℝ, known as shearing transformation.

2. In dimension = 1, consider the group SIM(2) given by

ℎ+(1 + ), B , â, = Û (1 + )ãäR 0(1 + )(Σ)ZRãäR f g ãäRÜ , > −1, B ∈ ℝ, â ∈ [0,2¶) (36)


Where (ã)äR = 9 'e â sin â− sin â cos â= )& (Σ)ZR =iB BB −Bj. This subgroup of Sp(2, ℝ) is also reproducing

[1, 3] and its semidirect structure are Σ ⋊ D, with Σ =

(Ú)ZR, B ∈ ℝ and

D = (1 + ) ãäR, > −1, â ∈ [0, 2π) ∈ ℝ. In this case,

we have

θ ((1 + ) ãäR)B = (() (1 + ) ãäR)YB, > −1, â ∈

[0, 2π), B ∈ ℝ.

The SIM(2) group is named so because it is isomorphic to

the group of similitude transformations of the plane. It is one

in the family of group(),3 introduced and studied in [3,

Section 6]. They display an analogous semidirect product

structure, and they all belong to ℰ.

4.2. The Mapping é

Much of the analysis of the metaplectic representation on a

group in the class Ԑ originates from the properties of a

fundamental quadratic mappings of ℝ() , whose basic

properties are described in the next proposition (see. e.g.,

[11]).

Proposition 10 There exists quadratic mapping Φ: ℝ() → ℝ(), that satisfies

(Ú) ∑ ?@k@ , ∑ ?@k@ $ = −2 1 + , Φ(∑ ?@k@ )$ (37)

Φ (a−1(∑ ?@k@ )) = Ý() (a)Φ(∑ ?@k@ ), (38)

for every ∑ ?@k@ ∈ ℝ() and every a ∈ D. The mapping θ is

defined in (33).

Proof. We shall compute explicitly the mapping Φ and

prove (37) and (38). Select a basises

12 of ℝ() and put (Ú)2 = j (12 ). Thus, if (1+) = ∑ (1 + )2()2 (12 ), then(Ú)=∑ (1 + )2()2 (Ú)2 and

(Ú)(V ?@k )@ , V ?@k@ $ = V (1 + )2 (Ú)2 V ?@k@ , V ?@k@ $()2 = −2 1 + , Φ UV ?@k@ X$

where

Φ+∑ ?@k@ , = +Φ(∑ ?@k@ ), ⋯ , Φ()(∑ ?@k@ ),, ºtlℎ Φ+∑ ?@k@ , = Y (Ú) (∑ ?@k )@ , ∑ ?@k@ $ (39)

This establishes (37). Finally, using (33) and (37), obtain

-21 + , Φ()Y+∑ ?@k@ ,)$ = (Ú))Y(∑ ?@k@ ), )Y(∑ ?@k )@ $ = )() (Ú)Ø(Å)()(∑ ?@k@ ), )Y(∑ ?@k )@ $ = (Ú)Ø(Å)()(V ?@k )@ , ))Y(V ?@k )@ $ = (Ú)Ø(Å)() V ?@k ,@ (V ?@k@ )$ = -2Ý())(1 + ), Φ(∑ ?@k@ )$ = −21 + , Ý())Φ(∑ ?@k@ )() $

for every (1+) ∈ ℝ(), hence equality (38). ∎

Would expect that the reproducing formula coming from

the metaplectic representation of H in (34) is somehow

equivalent to the wavelet case (25). This is what we are going

to exhibit.

The approach gives a general criterion for reproducibility

which contains those in [3, 4].

Let Φ: ℝ()→ ℝ() be quadratic mapping and assume

that 8ë, the Jacobians of Φ, does not vanish identically. Set

S =∑ ?@k@ ∈ ℝ(): 8ë(∑ ?@k@ ) = 0.

Notice that, if (B)~ ∈ ℝ() \ Φ (S) and (∑ ?@k@ )~ ∈ΦY(B)~ , then (∑ ?@k@ )~ ∉ S. Hence, by the local

invertibility theorem there exists an open neighborhoods

of (∑ ?@@ )~ and an open neighborhood B of (B)~ such that Φ|_R : → B is diffeomorphism. The local invertibility

theorem in o also tells us that in a neighborhoods of

∑ ?@k

@ ~ in o() there are no other solutions of Φ(∑ ?@k@ ) = (B)~ . Hence, being the solutions of Φ ( ∑ ?@k@ ) = (B)~

isolated, by the Bezout theorem they are at most 2(). To sum up, for any open set X ⊂ ℝì contained in Φ(ℝ()) \ Φ(S) such that the cardinality of ΦY (B ) are

locally constant for B ∈ X, the mapping Φ induce a

surjective finite-sheeted coverings

Φ: ΦY(X) → X

(observe that ΦY(X) is also open). In other terms, every (B)~ ∈ X has an open neighborhood B such that ΦY (B)

=í ( ) , k ≤ 2() , ( ) ⊂ ℝ( \ S,( ) open and ( ) ∩ ( )2 = Φ if i ≠ j, and Φ : ( ) → B is

diffeomorphism.

Remark. It could be useful to observe that the assumption

that the cardinality of ΦY(B) for B ∈ î is locally constant

(constant if X is connected) is in fact equivalent to requiring

the more easy to check hypothesis that the map Φ: ΦY(X) →î are propers, i.e. for every compact Ã ⊂ î , ΦY (K) is

compacts.

Indeed, assume that the cardinality of ΦY (B) for B ∈X

is locally constant. Then, as saw, Φ: ΦY(X) → X is finite-

sheeted covering, and therefore it is clear that ΦY (K) are


compact subset, when K is contained in one of the open

subset B above. In the case of a general compact K ⊂ X, let (∑ ?@k@ )ì ∈ ΦY (K) be sequence. Possibly after replacing (∑ ?@k@ )ì with subsequence, Φ(∑ ?@k@ )ì will converge to an

element Bj¡ ∈ K, so that it must belong to a compact subset ÃF

contained in an a small neighborhood B of Bj¡ . Hence (∑ ?@k@ )ì ∈ ΦY(¼F ), which is compact by what we have just

observed, so that still a sequence of (∑ ?@k@ )ì should

converge to an element (∑ ?@j@ ). By continuity, Φ (∑ ?@j@ ) = Bj¡ , so that (∑ ?@j@ ) ∈ ΦY(K).

Viceversa, suppose that the map Φ : ΦY (X) → X is

proper. Take Bj¡ ∈ X and let ΦY(Bj¡ ) = (∑ ?@j@),..,(∑ ?@j@). Already know that, for

convenient pairwise disjoint open neighborhoods ( ), j =

1,..., k of (∑ ?j@ )@ and for an open neighborhood B of Bj¡ , Φ : ( ) → B is diffeomorphism, so that every Bj

sufficiently close to Bj¡ has at least k pre-images, each

contained in one of the ( ) ’s. Suppose, by contradiction,

that there exists sequence (Bj)ì → (Bj¡ ), with each (B)ì having a further pre-images (∑ x@j@ )ì; hence (∑ x@j@ )ì ∉ í ( ) . By the hypothesis of

properness, one subsequences of (∑ ?@@ )ì must converge to

an elements (∑ ?@j@ ) ∉ í ( ) . By continuity we have Φ (∑ ?@j@ ) = Bj¡ , so that (∑ ?@j@ ) ∈ ΦY ( Bj¡ ) ⊂

í ( ) . which are contradictions.

With the notation above, we have the following result (see.

e.g., [11]).

Proposition 11. Suppose X ⊂ ℝ() is an open, simply

connected set, contained in Φ(ℝ() ) \ Φ(S) such that the

cardinality of ΦY(B) are constants for B ∈ X. Then there

exists an integer

k ∈ 1,..., 2() and there exist open, connected, and

pair-wise disjoint sets

Y1,..., ï ⊂ ℝ() \ S such that ΦY(X) = í ï ., and ∅| ñR : ï→ X is diffeomorphism.

Proof. Since Φ: ΦY(X) → X are covering and X is open

(hence locally path-connected) and simply connected, it

follows that the covering is trivial: i.e., there exists

homeomorphism Ψ: X × ΦY(B)~ → ΦY(X), (B)~ being

all fixed point in X. Since the cardinality of ΦY(B)~ are at

most 2(), we have

ΦY(B)~= (∑ ?@@ ),..., (∑ ?@@ ) ⊂ ℝ() \ S, k ≤ 2() and the desired result follows by taking ï = Ψ (X ×

∑ x@j@ ). ∎

Observe that, since Φ is even, if Φ is diffeomorphism from ï onto X, then it is diffeomorphism from -ï onto X. Hence

the number k of ï is even.

Examples. 1. In dimension = 1, consider the mapping Φ,

related to the TDS group in (35) and defined by (see [3,

(5.15)])

Φ U(V ?@@ ), (V ?@@ )X = U−(V ?@@ )(V ?@@ ), − (∑ ?@@ )2 X

Here S = f+∑ ?@@ ,, 0g , +∑ ?@@ , ∈ ℝ , Φ (S) = (0, 0) and Φ (ℝ) = ℝ × ℝY ∪ (0, 0).

If we define X =ℝ× ℝY, then X is open and simply connected and

ΦY (X) = (ï ∪ ï)

where

ï= ÷((∑ ?@@ ), (∑ ?@@ )) ∈ ℝ: (∑ ?@@ ) > 0ø,

ï = −Y1= ÷((∑ ?@@ ), (∑ ?@@ )) ∈ ℝ: (∑ ?@@ ) < 0ø.

2. In dimension = 1, consider the mapping Φ, related to the SIM(2) group in (36) and defined by (see [3, (5.11)])

Φ U(V ?@@ ), (V ?@@ )X = U(∑ ?@@ ) − (∑ ?@@ )2 , −(V ?@@ )(V ?@@ )X

Here S = (0, 0), Φ(S) = (0, 0) and Φ(ℝ) = ℝ. If we define

X = ℝ()\ ÷((∑ ?@@ ), 0), (∑ ?@@ ) ≤ 0 ø, then X is open and simply connected and

ΦY(X) = (Y1∪ Y2)

where

Y1 = ÷+(∑ ?@@ ), (∑ ?@@ ), ∈ ℝ: (∑ ?@@ ) > 0ø,

Y2 = −Y1= ÷+(∑ ?@@ ), (∑ ?@@ ), ∈ ℝ: (∑ ?@@ ) < 0ø.


3. In dimension > 1, consider the mapping Ψ(), related to the group

F = ℍ! ⋊ U () ⊂Sp(1 + , ℝ), with ℍ! the Heisenberg group extended by the usual

1-dimensional homogeneous dilations, studied in [4] and defined by (see [4, (20)])

Ψ() ÀUV ?@@ X’ , (V ?@@ )()Á= ÀUV ?@@ X’ , (V ?@@ )() − 12 (V ?@@ )() (1 + ), 12 (V ?@@ )() Á , ÀUV ?@@ X’ , (V ?@@ )()Á∈ ℝ × ℝ,

Where (1+) ∈ ℝ is fixed. Here S = (+∑ ?@@ ,’, 0), +∑ ?@@ ,’ ∈ ℝ, Φ (S) = (0, 0) and Φ(ℝ()) = ℝ×ℝ ∪ (0, 0). If we define X = ℝ

× ℝ+, then X is open and simply connected and

ΦY(X) = (Y1∪ Y2)

where

Y1 =(+∑ ?@@ ,’,(∑ x@j@ )()) ∈ ℝ × ℝ: (∑ ?@@ )() > 0,

Y2 = −Y1 = (+∑ ?@@ ,’,(∑ x@j@ )()) ∈ ℝ × ℝ: (∑ ?@@ )() < 0

4. In dimension = 1, consider the mapping Φ, related to the TDH group, defined in the subsequent (49), given by (see (52))

Φ U(V ?@@ ), (V ?@@ )X = U− 12 U(V ?@@ ) + (V ?@@ )X , −(V ?@@ )(V ?@@ )X

We have

S =((∑ ?@@ ), (∑ ?@@ )) ∈ ℝ: (∑ ?@@ ) = ± (∑ ?@@ ),

Φ(S) =((∑ ?@@ ), (∑ ?@@ )) ∈ ℝ: (∑ ?@@ )= ±(∑ ?@@ ), (∑ ?@@ ) ≤ 0

If we set

X =(u,û) ∈ ℝY × ℝ: u2 − (û) > 0, (40)

then

Φ(ℝ2)= X ∪ (0, 0).

In this case k = 2 = 4 and

ΦY(X) = Y1∪ Y2∪Y3∪Y4,

where

Y1 = ((∑ ?@@ ), (∑ ?@@ )) ∈ ℝY× ℝ: (∑ ?@@ ) − (∑ ?@@ ) > 0,

Y2 = −Y1 = ((∑ ?@@ ), (∑ ?@@ )) ∈ ℝ+× ℝ: (∑ ?@@ ) − (∑ ?@@ )> 0, (41)

Y3=((∑ ?@@ ), (∑ ?@@ )) ∈ ℝ ×ℝY: (∑ ?@@ ) − (∑ ?@@ ) < 0,

Y4 = −Y3 = ((∑ ?@@ ), (∑ ?@@ )) ∈ ℝ ×ℝ+:(∑ ?@@ ) − (∑ ?@@ )< 0 (42)

Lemma 12. Let Φ, ï, X be as in Proposition 11. If h ∈ o~ (ï), then

∑ ü ℎ+∑ ?@@ ,1S2,ëf∑ MNRN g$&(∑ ?@@ýR )ü &(1 + ) = ∑ ℎ(∑ ?@@ ) þ(∑ MNR)N(∑ MNRN )ýRℝ(wxy) (43)


where 8ë(∑ ?@@ ) are the Jacobian of Φ at ∑ ?@@ .

Proof. Recall that ï ⊂ ℝ() \ S, so that 8ë(∑ ?@@ ) ≠ 0 on ï and Φ:=Φ|ñRare diffeomorphism from ï onto X. This let us

make the change of variables Φ(∑ ?@@ ) = u and use Plancherel’s formula:

v V v ℎ UV ?@@ X 1S2(),ëf∑ MNRN g$& UV ?@@ XýR

ℝ(wxy)

&(1 + )

= v v ℎ ÀV ΦY(Ó) Á 1S2(),$ 8ëRÖw(Ó) &Ó

ℝ(wxy)&(1 + )

= v V v îî(Ó)ℎ fΦY(Ó)g 1S2(),$ 8ëRÖw(Ó) &Óℝ(wxy) ℝ(wxy) &(1 + ) = v V îî(Ó)ℎ fΦY(Ó)g 8ëRÖw(Ó)ℝ(wxy) &Ó

= v V ℎ(ΦY(Ó))8ëRÖw(Ó) &Ó = v V ℎ(V ?@@ ) ýR

&(∑ ?@@ )|8ëî|

where in the last row have performed the change of variables ΦYÓ =∑ ?@

@ . Observe that, since the supp h is a compact set

contained in the open sets ï , there exist two constants 0 < < C, such that < ∑ 8ë(∑ ?@@ ) < C on supph and the last

integral are well-defined. ∎

We have all the pieces in place to provide the reproducing condition on the sequence of wavelet 5 which guarantees the

reproducibility of the group H (see. e.g., [11]).

Theorem 13. Let H = Σ ⋊ D ≅ ℝ() ⋊D be as at the beginning of this section and let X, ï be as in Proposition 11. Then,

the identity

∑ // = ∑ , +ℎ(1 + , )),5$&ℎ(1 + , ))% (44)

holds for every ∈ o~ (ï) if and only if 5 satisfies the condition

∑ 5()Y+∑ x@@ ,)É |de l θaa|Y&) ∑ ü8fë∑ sNRN gü , ). 1. ∑ x@@ ∈ ï (45)

Moreover, (44) holds for every ∈ o~(ï ∪ (−ï)) if and only if 5 satisfies the following two conditions:

∑ 5()Y+∑ x@@ ,)É |de l θ(a)a|Y&) ∑ 5−)Y+∑ x@

@ ,É |de l θ(a)a|Y&) ∑ 8ë∑ sN

RN (46)

For a.e. ∑ x@

@ ï, and

∑ 5−)Y+∑ x@

@ ,5j)Y∑ x@j@

É |de l θ(a)a|Y&) 0, ). 1. ∑ x@@ ∈ ï (47)

Proof. The left Haar measures on H is given by

dh(1+, a) = d(1+)|de l Ý)|Yda

and the metaplectic representation on H in (34), let us write, for every ∈ o~ (ï ∪ (−ï)),

v V . fℎ((1 + ). ))g 5$ &ℎ(1 + , ))%

= v v V v UV ?@@ X 1YS2Þ(wxy) ∑ MNRN ,∑ MNRN $(de l ))Y ⁄ 5jU)Y(V ?@j )@ X

ℝ(wxy) &(V ?@)@ ∙ℝ(wxy)É

(&(1 + )|de l Ý()))|Y&)


= v v V v UV ?@@ X 1S2,ëf∑ MNRN g$5jU)Y(V ?@j@ ))X

ℝ(wxy) &(V ?@)@ ∙ℝ(wxy)É

(&(1 + )|de l Ý()))|Y&))

= ∑ ∙ ü n+∑ ?@@ ,5j+)Y(∑ ?@j@ ), + +−(∑ ?@j )@ ,5

j+−)Y(∑ ?@j@ ),p 1S2,ëf∑ MNjN g$&(∑ ?@j )@ýR ü

ℝ(wxy)É ∙ (&(1 + )|de l Ý()))|Y&)) (48)

where the last equality are due to the even property of Φ. Sets h(∑ ?@j@ ): =+∑ ?@j@ ,5j+)Y ∑ ?@j@ ,

+ +− ∑ ?@j@ ,5j+−)Y ∑ ?@j@ , and apply lemma 12, so that

v V, +ℎ+(1 + ). ),5,$ &ℎ(1 + , ))

%= v v ÀV UV ?@j@ X 5

jU)Y(V ?@j@ )X

+ U− V ?@j@ X 5jU−)Y( V ?@j@ )X Á

ýRÉ

+ À2ℛ1 U U(V ?@j )@ X 5jU)Y(V ?@j )@ XX

jU−(V ?@j@ )X 5k U−)Y(V ?@j@ )XÁ &(∑ ?@j )@8ë(∑ MNjN ) |&1l Ý ()))|Y &)

Suppose at first that satisfies the additional properties: +∑ ?@j@ , = 0 on −ï, then

v V. +ℎ((1 + ). )),5$% &ℎ(1 + , ))

= v V UV ?@j@ X ýR Àv 5 U)Y V ?@j@ X |&1l Ý ()))|YÉ &)Á & ∑ ?@j@8ë(∑ MNjN )

so that % ∑, (h(1 + ε, a))# $ &ℎ(1+, a) = ∑ // if and only if (45) holds.

If, instead, (∑ ?@j@ ) = 0 on ï, the equality % ∑ , (h(1 + ε, a))# $ dh(1+, a) = ∑// holds true ifnd only if the

second and the last expression in (46) are equal.

Finally, taking ∈ o~(ï ∪ (−ï)), such that (∑ ?@j@ )(− ∑ ?@j@ ) are real-valued, purely imaginary-valued, respectively,

we have

v V. +ℎ((1 + ). )),#$&ℎ(1 + , )) =% V//

if and only if both conditions (46) and (47) are fulfilled. ∎

Corollary 14. Theorem 13 still holds if the assumptions ∈ o~(ï) or ∈ o~(ï ∪ (−ï)) are replaced by ∈ (ï), or ∈ (ï ∪ (−ï)), respectively.

Proof. It follows by the density of o~ (ï) and o~ (ï ∪ (−ï)) in (ï) and (ï ∪ (−ï)), respectively. ∎

4.3. The Case = ℝ() ⋊

We shall exhibit that for subgroups of the kind = ℝ() ⋊ Ù, with N defined in (30), the series formula (1) ever fails. The

product laws are given by

ℎ(1 + , )ℎ G1 + , H = ℎ UD + + (1 + ) I , X

First of all, in the semidirect product above, we are dealing with the case


ℝ() ≅ 0 × ℝ() ⊂ ℝ(). The choice ℝ() ≅ ℝ()× 0 reduces ℝ() ⋊ N to ℝ(), since it forces N to be I.

In case, for (1+),f g ∈ ℝ(), = n1 0 1p, with () = ∈ P(1 + , ℝ)(), the action on ℝ() is

D + + (1 + ) I = n 01 + p + n 1 0 1p Û 01 + Ü = Û 0 + 2 + 1 Ü

and the product law becomes

h(1+, )(ℎ) Gf g , jF H = hf , + jF g.

The extended metaplectic representations on H is given by

!(ℎ(1 + , n) ) (1 + ) = T(0, 1+) (n)(1 + ) = ±P()12SR(),$(1+)

The right-hand side of (1) has the series form

v V, !(ℎ) 5$%

& ℎ = v v V ℱ f12SR ∙,∙$5j¡ g (1 + )

ℝ(wxy) &(1 + )& ()

= v v V f12SR(),$(1 + )5j(1 + )gℝ(wxy)

&(1 + )& ()

= v V & () v (1 + )ℝ(wxy)

5(1 + ) &(1 + )

where used Plancherel’s formula, so that the last integral either vanishes or diverges.

5. New 2-Dimensional Reproducing Subgroups = ⋊

We shall construct two new examples of reproducing subgroups H = Σ ⋊ D, in dimension = 1. To prove their

reproducibility, we shall apply the theory developed (see, e.g., [11]).

5.1. The Group

Consider the 4-dimensional group:

L = ℎ ÀU(V x@@ ), BX , (e , 1 + )Á = 1YdR (1 + ) 01dR L U(V x@@ ), BX (1 + ) 1dR +−(1 + ),:

e , (1 + ), ∑ x@@ , B ∈ ℝ ⊂ (2, ℝ) (49)

with the hyperbolic matrix H (1 + ) given by

(1 + ) = 9cosh(1 + ) sinh(1 + )sinh(1 + ) cosh(1 + )= , (1 + ) ∈ ℝ (50)

whereas the symmetric matrices T(∑ x@@ , B) displays the entries

L+∑ ?@j@ , B, = i∑ ?@j@ BB ∑ ?@j@ j , ∑ ?@j@ , B ∈ ℝ (51)

The semidirect structure H = ℝ2 ⋊ D is clear:


1YdR (1 + ) 01dR L(V ?@j@ , B)(1 + ) 1dR (−(1 + )) = 1 0L UV ?@j@ , BX 1 i1YdR (1 + ) 00 1dR (−(1 + ))j

Proposition 15. The subgroups TDH of Sp(2,ℝ) satisfy the following properties:

(a) The product law in TDH are explicitly given by:

h (e, 1 + ,E)ℎ ((ej)F , f g,EjF ) = ℎ (e +(ej)F , f g, E+ 1dR H (−2(1 + ))EjF ),

E,EjF ∈ ℝ, 1 + ,f g , e,(ej)F ∈ ℝ

(b) The left Haar measure on TDH is dh (e, 1 + ,E) = 1YdRde d(1 + ) dE.

(c) The mapping Φ in (39) is explicitly given by

Φ+∑ x@@ , = f− +(∑ ?@j@ ) + (B),, −(∑ ?@j )@ Bg (52)

and has Jacobians 8ë(∑ ?@@ , B) = −((∑ ?@j@ ) − (B)). Observe that Φ(ℝ) = X ∪ (0, 0), where theopen set X is defined in

(40).

(d) The restriction of the metaplectic representations to TDH is given by:

(h (e,(1 + )), (∑ ?@j@ , B))(u) = ±1dR1S2(∑ MNjN ,ZR)),$(1dR(−(1 + ))Ó), ∈ (ℝ) (53)

(e) The group homomorphism θ in (29) is

Ý(1YdRH (1 + )) = 1dR

H (−2(1 + )) = f1YdRH (1 + )gY = (1YdRH (1 + ))() Y (54)

since the matrix a = a(e, 1 + ) = 1YdRH(1 + ) is symmetric. Hence θ is the same homomorphism encountered in the TDS(2)

and SIM(2) group cases.

The reproducibility of the group TDH are then a mere application of Theorem 13, with ΦY(X) = Y1∪ Y2∪ Y3∪Y4, and X defined in (40), Y1, Y2 = −Y1 defined in (41) and Y3, Y4 = −Y3 defined in (42).

Theorem 16. The subgroup TDH is reproducing for (Y1 ∪ (−Y1)). Moreover, 5 ∈ (Y1∪ (−Y1)) are reproducing sequence of functions for TDH if and only if

∑ 5(Ó, û)ýw þþR+Y(R), = ∑ 5(−Ó, −û)ýw þþR

+Y(R), = 1 (55)

and

∑ 5(−Ó, −û)5j(Ó, ûj)ýw þþj+Y(R), = 0 (56)

Similarly, TDH is reproducing for (Y3∪ (−Y3)), and 5 ∈ (Y3∪ (−Y3)) are reproducing sequence of functions if fulfills

(55) and (55) with Y3 in place of Y1.

Proof. Use Theorem 13 and translate conditions (46) and (47) into this context. The automorphism θ is computed in (54),

hence

|det θ (a)a |= ∑ üdet Gf1YdR H(1 + )gY 1YdR H (1 + )Hü = ∑ det(1dR H (−(1 + )) Y =∑ 1YdR .

Consider the case Φ = Φ|ñw . The first condition in (46) reads in this frame work as

v V 5 U1dR +−(1 + ),(()UV x@k@ , BX)X ℝ(wxy)

1YdR &e&(1 + )

= V(V x@k@ ) − (B) ). 1. (V x@k@ , B) ∈ ï


Performing the change of variables 1dRH (−(1 + )) (∑ x@k@ , B) ()

= (u, ûj) (), get 1dR

((∑ x@k@ ) −(B)) = u2 − (ûj). Here ded(1 + )=

(j) dudûj

Hence, the previous integral coincides with the first on the left-hand side of (55). The other cases are analogous. ∎

5.2. The TDW Group

The TDW group arises by tensor-product of 1-dimensional wavelets and is defined as follows.

= ℎ ÀUV x@@ , BX , (e , 1 + )Á =ÍÎÎÎÎÎÏ

ÍÎÎÎÎÏ 1dR0 01() 001dR V x@k@ 0 1YdR

0 1()B 00001Y()ÐÑ

ÑÑÑÒ

: e , (1 + ), V x@@ , B ∈ ℝÐÑÑÑÑÑÒ

⊂ e(2, ℝ)

The TDW group enjoys the following properties:

(i) If we sets a (e, 1 + ) = 91dR 00 1()= the product law in H are explicitly given by:

h (e, 1 + ,E)h((ej)F , f g, (EjF )) = h(e +(ej)F , f g , Ej+ a(e , 1 + )YEjF ), EEjF ∈ ℝ,

(1 + ), f g , e, (ej)F ∈ ℝ. Hence the automorphism θ is θ (a(e, 1 + )) = a(e , 1 + )Y (notice that (e , 1 + ) are symmetric).

(ii) The mapping Φ in (39) is given by

Φ UV ?@j@ X = − 12 U(V ?@j@ ), (B)X

and has Jacobians 8ë (∑ ?@j@ , B) = ∑ x@@ B, so that 8ë(∑ ?@j@ , B) = 0 on the set

S = (∑ ?@j@ , B): ∑ ?@j@ = 0 ∨ B = 0 and Φ(S) = (∑ ?@j@ , 0), ∑ ?@j@ ≤ 0 ∪ (0, B), B ≤ 0. Moreover, Φ(ℝ2) = (∑ ?@j@ , B): ∑ ?@j@ ≤ 0, B ≤ 0.

In this case, we have

X = ℝ − × ℝ−,ΦY(X) = Y1∪ (−Y1) ∪ Y2 ∪ (−Y2),

where

Y1 =÷(∑ ?@j@ , B) ∈ ℝ × ℝø, Y2 = ÷(∑ ?@j@ , B) ∈ ℝY × ℝø.

(iii) The restriction of the metaplectic representation to is given by:

μ fℎ+(e , 1 + ), (∑ ?@j@ , B)) ,g (u, û) = ±1YdR ⁄ 1S2 ∑ sNRN 1Y() ⁄ 1S2ZR(R) f1YdRÓ, 1Y() ûg (57)

Theorem 13 rephrased for the TDW group is as follows.

Theorem 17. The subgroup TDW is reproducing on (Y1∪ (−Y1)). Sequence of functions 5 ∈ (Y1∪ (−Y1)) are reproducing sequence of functions if and only if

v V5(Ó, û)ýw

&Ó &ûÓ(û) = v V5(−Ó, −û) &Ó &ûÓ(û) = 1ýw

∑ 5(−Ó, −û)5j(Ó, ûj) þ þj(j) = 0ýw (58)

and, similarly, TDW is reproducing on (Y2∪ (−Y2)).

Recall the reproducing subgroup ⊂ ℝ ⋊ SL(2, ℝ) given by

= ÊGn00p , n 1 0) + 1p 9)Y ⁄ 00 ) ⁄ = , > −), () + ) ∈ ℝHË

reproducing sequence of functions (5) are reproducing for if and only if (5) ∈ (ℝ) and


v V (5)(V ?@j@ )

~

&(∑ ?@j )@(∑ ?@j@ ) = v V (5)(− V ?@j@ ) &(∑ ?@j )@(∑ ?@j@ ) = 12

~,

v V+5, UV x@k@ X +5j,(− V ?@j@ )

~&(∑ ?@j )@(∑ ?@j@ ) = 0

It is then clear that every sequence of functions 5(u, ûj) =

4 (5) (u) (5) ( ûj ) fulfills (58), so that a reproducing

sequence of function is obtained by a tensor product of two

1-dimensional the sequence of wavelets.

Acknowledgements

We would like to thank our colleagues for interesting

discussions and helpful ideas. Moreover, we thank the

anonymous referee for suggestions to improve the paper.

References

[1] S. T. Ali, Antonine, J. P. Gazeau, Wavelets and Their Generalization, Springer-Verlag, New York, 2000.

[2] E. J. Candes, D. L. Donoho, L. David, Curvilinear integrals, J. Approx. Theory 113 (1) (2001) 59-90.

[3] E. Cordero, F. De Mari, K. Nowak, and A. Tabacco, Analytic features of reproducing groups for the metaplectic representation, J. Fourier Anal. Appl 12 (3) (2006) 157-180.

[4] E. Cordero, F. DeMari, K. Nowak, A. Tabacco, A dimensional bound for reproducing subgroups of the symplectic group, Mth. Nachr. 283 (7) (2010) 1-12.

[5] S. Dalke, G. Kutyniok, G. Steidl, G. Teschke, Shearletcoorbit spaces and associated Banach frames, Appl. Comut. Harmon. Anal. 27 (2) (2009) 195-214.

[6] H. G. Feichtinger, M. Haewinkel, N. Kaiblinger, E. Matusiak, M. Neuhauser, Metaplectic operator on ℂ, Q, J. Math. 59 (1) (2008) 15-28.

[7] K. Grochenig, Foundation of time-Frequency Analysis, Birkhiuser, Boston, 2001.

[8] K. Guo, D. Labate, Spare shearlet representation of Fourier integral operators, Electron. Res. Announc. Math. Sci, 14 (2007) 7-19.

[9] R. S. Laugesen, N. Weaver, G. L. Wilson, A characterization of the higher dimensional groups associated with continuous wavelets, J. Geom. Anal. 12 (1) (2002) 89-102.

[10] G. Weiss, E. N. Wilson, The Mathematical theory of wavelets, in: Twentieth Century Harmonic Analysis – A Celebration, II Ciocco, 2000, in: NATO Sci. Ser. II Math. Phy. Chem., vol. 33, 2001, pp. 329-366.

[11] E. Cordero, A. Tabacco, Triagular subgroups of Sp(d, ℝ) and reproducing formulae, Journal of Functional Analysis 264 (2013) 2034-2058. http://dx.doi.org/10.1016/j.jfa.2013.02.004

article.aascit.orgarticle.aascit.org/file/pdf/9210811.pdf · international journal of mathematical...

Documents