Multi-Dimensional Stockwell Transforms and Applications

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Multi-Dimensional Stockwell Transforms and Applications

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Università degli Studi di Torino Dipartimento di Matematica

Scuola di Dottorato in Scienza ed Alta Tecnologia

Ciclo XXVI

Multi-Dimensional Stockwell

Transforms and Applications

Luigi Riba

Relatore: Prof. Man Wah Wong (York University, Toronto)

Coordinatore del Dottorato: Prof. Ezio Venturino Anni Accademici: 2011-2013

Università degli Studi di Torino

Scuola di Dottorato in Scienza ed Alta Tecnologia

Tesi di Dottorato di Ricerca in Scienza ed Alta Tecnologia Indirizzo: Matematica

MULTI-DIMENSIONAL STOCKWELLL

TRANSFORMS AND APPLICATIONS

Luigi Riba

Relatore: Prof. Man Wah Wong (York University, Toronto)

Contents

1 Introduction 2

2 Notation 8

3 Group representation theory 11

3.1 Localization operators and homogeneous spaces . . . 11

3.2 Affine group . . . 13

3.3 Similitude group . . . 14

3.4 Weyl-Heisenberg group . . . 16

3.5 Reduced Heisenberg group . . . 17

3.6 Affine Heisenberg group . . . 19

4 Stockwell transforms 22 4.1 Stockwell transforms and affine Heisenberg group . . . 26

4.2 Gabor transforms . . . 26

4.3 Moritoh wavelet transforms . . . 28

5 Continuous inversion formulas 30 5.1 Constant matrices . . . 32

5.2 Diagonal matrices . . . 33

5.3 Rotation matrices in dimension n“1, 2, 4, 8 . . . 35

5.4 A unification . . . 42

6 Localization operators 46 7 Instantaneous frequency 50 7.1 Phase of the multi-dimensional Stockwell transform and instantaneous fre-quency . . . 50

7.2 Examples . . . 60

7.2.1 1-dimensional case . . . 60

7.2.2 Diagonal matrices . . . 61

7.2.3 Rotation matrices in dimension n“1, 2, 4, 8 . . . 61

1 Introduction

Let f be a signal in L2_p

Rq. Then we can recover frequency information contained in the signal using the Fourier transform F f given by

pF fq pξq “ p2πq´1{2 ż

R

e´i xξf pxqd x , ξPR.

Sometimes it is not enough to have information regarding the frequency content of the sig-nal f . In these cases it is useful to look at the Gabor transform V_ϕf of the signal with respect

to a certain windowϕ in L2pRqgiven by

`

V_ϕf˘pb ,ξq “ p2πq´1{2 ż

R

e´i xξf pxqϕpx´bqd x

for all b ,ξPR. We can notice that

`

V_ϕf˘pb ,ξq “ p2πq´1{2`f , M_ξT´bϕ

˘

L2_pRq

for all b ,ξPR, wherep,q_L2_pRqis the inner product in L2pRq, Mξand T_´b are the modulation

operator and the translation operator given by

`

M_ξf˘pxq “ei xξf pxq, xPR, and

pT´b fq pxq “f px´bq, xPR,

for all measurable functions f on R.

The usefulness of the Gabor transforms in signal analysis is enhanced by the following res-olution of the identity formula, which allows the reconstruction of a signal from its Gabor transform.

Theorem 1.1. Suppose that  ϕ    

L2_pRq“1, where|| ||L2pRqis the norm in L2pRq. Then for all

f and g in L2pRq, pf , gqL2_pRq“ ż R ż R ` V<sub>ϕ</sub>f˘pb ,ξq`V_ϕg˘pb ,ξqd b dξ. (1.1)

Another way of looking at Theorem 1.1 is that for all f in L2_p Rq, f “ p2πq´1 ż R ż R pf , M_ξT_´bϕqL2_pRn_qM_ξT_´bϕ d b d ξ.

The formula in Theorem 1.1 is also known as a continuous inversion formula for the Gabor transform and it is related to the representations of the Heinsenberg group H.

We can use (1.1) to define localization operators L_F,ϕwith symbol F PLp_p

Rqand window

ϕ associated to the Gabor transform by

` LF,ϕf , g ˘ L2<sub>pRq</sub>“ ż R ż R Fpb ,ξq`V_ϕf˘pb ,ξq`V_ϕg˘pb ,ξqd b dξ.

These operators are studied in detail in[32, 33].

In signal analysis, the term`V_ϕf˘pb ,ξqgives the time-frequency content of a signal f at time b and frequencyξ by placing the window ϕ at time b .

It is important to point out that we can extend the Gabor transform here defined to the multi-dimensional case. In fact it is sufficient to look at

` V<sub>ϕ</sub>f˘pb ,ξq “ p2πq´n {2`f , M_ξT´bϕ ˘ L2_pRn_q “ p2πq´n {2 ż R e´i xξ_f pxqϕpx´bqd x , b ,ξPRn.

The drawback here is that a window of fixed width is used for all time b . It is more accurate and desirable if we can have an adaptive window that gives a wide window for low frequency and a narrow window for high frequency. That this can be done comes from familiarity with the wavelet transform that we now recall.

This can be done by looking at the wavelet transform W_ϕf of the signal f PL2_p

Rqwith respect to a certain windowϕ in L2pRqgiven by

` W_ϕf˘pb , aq “a´1{2 ż R f pxqϕ ˆ x´b a ˙ d x

for all bPR, aPR`. We can notice that

`

W_ϕf˘pb , aq “ pf , T´bD2,aϕqL2_pRn_q

for all bPR, aPR`, where D2,a is the dilation operator given by

pD2,a fq pxq “a´1{2f

´x

a

¯

for all measurable functions f on R.

The nucleus of the analysis of the wavelet transform is the following resolution of the iden-tity formula, which is a continuous inversion formula.

Theorem 1.2. Letϕ be a nonzero function in L2pRqsuch that

c_ϕ“2π

ż

R

|ϕˆpξq|2dξ

|ξ|ă 8. (1.2)

Then for all functions f and g in L2pRq,

pf , gqL2_pRq“ 1 c_ϕ ż R` ż R ` W_ϕf˘pb , aq`W_ϕg˘pb , aqd b d a a2 .

Notice that (1.2) is also known as admissibility condition for wavelets. A nonzero function

ϕPL2pRnqsatisfying (1.2) is called a mother affine wavelet. The adjective affine comes from the connection with the affine group A that is the underpinning of the wavelet transforms. See Chapter 18 of[33] in this connection.

A multi-dimensional version of the wavelet transforms has been introduced in[27] and is given by ` W_ϕf˘pb , a , Rq “ pf , T_´bD2,a RϕqL2_pRn_q “ 1 an {2 ż Rn f pxqϕ ˆ 1 aR ´1_px´bq ˙ d x

for allpb , a , Rq PRnˆR`ˆSOpn , Rq, where D2,a Ris the dilation operator given by

pD_{2,a R}fq pxq “a´n {2_f `

a´1_R´1_x˘

, xPRn, for all measurable functions f on Rn.

Now, letϕPL1_p

Rq XL2_p

Rq. Then, combining the merits of the Gabor transform and the wavelet transform, the Stockwell transform S_ϕf with windowϕ of a signal f in L2_p

Rqis defined by pS_ϕfqpb ,ξq “ p2πq´1{2|ξ| ż R e´i xξfpxqϕpξpx´bqqd x , (1.3)

for all bPR and ξPRzt0u. We note that for all f in L2_p

Rq, all b in R and all ξ in Rzt0u,

where the dilation operator D_1,1

ξ is defined by

pD_1,1

ξ fqpxq “ |ξ|fpξxq

for all x in R and all measurable functions f on R. Besides the modulation with respect to frequencyξ, a notable feature in the Stockwell transform is the normalizing factor in the dilation operator, which is|¨|in lieu of|¨|1{2as in the case of the wavelet transform, and is the mathematical underpinning of the absolutely referenced phase information in[31]. These features distinguish the Stockwell transform from the wavelet transform. Notwithstanding these differences, we have the following formula in[31] relating the Stockwell transform to the Morlet wavelet transform W_ψ.

Theorem 1.3. For all f in L2pRq,

pS_ϕfqpb ,ξq “ p2πq´1{2e´i bξ

b

|ξ|pΩ_ψfqpb , 1{ξq

for all bPR and ξPRzt0u, where

ψpxq “ei xϕpxq

for all x in R.

The Stockwell transform is closely related to the wave packet transform of Cordoba and Fef-ferman[10], which also involves a combination of translations, modulations and dilations. It should also be mentioned that transforms closely related to the wavelet transforms and the metaplectic representation abound and can be found in the monographs[14], and the works[9, 2].

Of particular importance in the Stockwell transform is the phase correction in the preced-ing formula given by e´i bξ_{, which is caused by the phase function e}´i xξ_{inside the integral}

defining the Stockwell transform. It is crucial to note that this function picks out the fre-quency to be localized, but is not translated with respect to time b as is always done for the Morlet wavelet transform[17]. To see the full significance of this, we note that in real-life applications, signals f and windowsϕ are real-valued functions. Therefore information about the phase argpS_ϕfqpb ,ξqof the Stockwell transformpS_ϕfqpb ,ξqat time b and fre-quencyξ comes from the term e´i xξ_{at time b “0. But in the case of the Morlet wavelet}

transform, the phase information is obtained by referencing the windowed signal f with re-spect to e´i px ´b qξ_{. This is precisely the absolutely referenced phase information in[13] and is}

responsible for the continuous inversion formula in Theorem 1.4 given later. Another point is that the Stockwell transform is reminiscent of the Morlet wavelet transform, but the ap-plicability of the computational techniques available for the Morlet wavelet transforms is undermined by the inversion a“1{ξ.

The Stockwell transform has recently been used in geophysics[13, 31] and in medical imag-ing[16, 35]. More recent applications in imaging are in [18, 19]. In view of its versatility, an attempt in understanding the mathematical underpinnings of the Stockwell transform is worthwhile. The following continuous inversion formula for the Stockwell transform can be found in[13] and [34] for the case when

ϕpxq “e´πx2 for all x in R. Theorem 1.4. LetϕPL1_p Rq XL2_p Rqbe such that ż R ϕpxqd x“1.

Then for all f in L2pRq,

f “F´1AS_ϕf ,

where F´1_{is the inverse Fourier transform and A is the time average operator given by}

pAFqpξq “ ż

R

Fpb ,ξqd b

for allξ in R and all measurable functions F on RˆR, provided that the integral exists. Another continuous inversion formula for the Stockwell transform akin to the continuous inversion formulas for the Gabor transform and the wavelet transform is given by the fol-lowing theorem. Theorem 1.5. LetϕPL2_p Rqbe such that c_ϕ“ ż R |ϕˆpξ´1q|2dξ |ξ|ă 8. (1.4)

Then for all f and g in L2pRq,

pf , gqL2_pRq“ 1 c_ϕ ż R ż R ` S<sub>ϕ</sub>f˘pb ,ξq`S_ϕg˘pb ,ξqd b dξ |ξ|.

Remark 1.6. Theorem 1.5 tells us that every signal f can be resconstructed from its

Stock-well spectrum by means of the formula

f “ 1 c_ϕ ż R ż R ` S_ϕf˘pb ,ξq ´ M_ξT_´bD_1,1 ξϕ ¯ d b dξ |ξ|.

Equation (1.4) means that ˆϕp´1q “0 whenever ˆϕ is continuous at´1. So, it is important to observe that the Gaussian window used exclusively for the Stockwell transform in the literature is not admissible.

The aim of this work is to provide a theoretical setting in which to study the Stockwell trans-forms. We introduce multi-dimensional Stockwell transforms and we give the correspond-ing continuous inversion formulas analogous to Theorem 1.4 and Theorem 1.5 for the 1-dimensional Stockwell transforms. We elucidate a connection between the affine Heisen-berg group AH and the multi-dimensional Stockwell transforms. Furthermore, we extend the results given in the one-dimensional case regarding the LppRnq-boundedness of the localization operators and the instantaneous frequency.

After Chapter 2 on notation, in Chapter 3 we give a brief introduction on group representa-tion theory and its relarepresenta-tions with time-frequency analysis. In particular we are interested in unitary irreducible and square integrable representations of groups or homogenous spaces. In this chapter we introduce and study the affine group A, the reduced Heisenberg group H and the affine Heisenberg group AH. In Chapter 4 we define the multi-dimensional Stock-well transform and we show its relations with the Gabor transform and the Moritoh wavelet transform. The role played by a certain section of the affine Heisenberg AH group in defin-ing the Stockwell transform is mentioned too. In Chapter 5 we prove continuous inversion formulas for multi-dimensional Stockwell transforms under different sets of hypotheses. Moreover, we provide a couple of interesting examples of Stockwell transforms that satisfy these set of hypotheses. The results in this chapter extend those in[24] and [29]. Chapter 6 is devoted to the study of the localization operators associated with multi-dimensional Stockwell transforms. We write an explicit relation between these operators and the Weyl transform. This chapter is an extension of the results given in[23]. In Chapter 7 we extend the results given in[19] for the 1-dimensional Stocwkell transform regarding the instanta-neous frequency of the signal.

2 Notation

Throughout all this paper we use the following operators on L2_p

Rnq. • Let ξPRn, then the modulation operator Mξis given by

`

M_ξf˘pxq “ei x ¨ξfpxq, @xPRn.

• Let bPRn, then the translation operator T´b is given by

pT´b fq pxq “fpx´bq, @xPRn.

• Let APGLpn , Rq, then the dilation operator Ds ,Ais given by

pDs ,Afq pxq “|det A|´1{sf

`

A´1x˘, @xPRn.

In the following we will often write DAinstead of D2,A.

Definition 2.1. We define the Fourier transform F f of the signal f by

pF fq pξq “ pf pξq “ p2πq´n {2 ż

Rn

e´i x ¨ξfpxqd x

for allξPRn.

Notice that F, M_ξ, T´b and DA are unitary operators on L2pRnq. They obey the following composition rules:

F M_ξ“T´ξF, F T´b “M´bF, F DA“DpA´1_qtF,

M_ξF“F T_ξ, M_ξT´b “ei b ¨ξT´bMξ, MξDA“DAMAt_ξ,

T_´bF“F Mb, T´bMξ“e´i b ¨ξMξT´b, T´bDA“DAT´A´1_b,

DAF“F DpA´1_qt, DAMξ“M_pA´1_qt_ξDA, DAT´b “T´AbDA.

Definition 2.2. We define the Gabor transform V_ϕf of the signal f with respect to the

win-dowϕ by

`

V_ϕf˘pb ,ξq “ p2πq´n {2`f , M_ξT´bϕ

˘

“ p2πq´n {2 ż Rn e´i x ¨ξ_f pxqϕpx´bqd x for allpb ,ξq PRnˆRn.

Definition 2.3. We define the wavelet transform W_ϕf of the signal f with respect to the

windowϕ by ` W<sub>ϕ</sub>f˘pb , a , Rq “ pf , T<sub>´b</sub>D2,a RϕqL2<sub>pR</sub>n<sub>q</sub> “ 1 an {2 ż Rn f pxqϕ ˆ 1 aR ´1<sub>px´bq</sub> ˙ d x for allpb , a , Rq PRnˆR`ˆSOpn , Rq.

Definition 2.4. Let R : RnQξÞÑRpξq “R_ξPSOpn , Rq, then we define the Moritoh wavelet transform W1

|ξ|R´1,ϕf of the signal f with respect to the windowϕ by

´ W1 |ξ|R´1,ϕf ¯ pb ,ξq “|ξ|n {2 ż Rn f pxqϕ`|ξ| Rξpx´bq˘d x “ ´ f , T_´bD_2,1 |ξ|R ´1 ξ ϕ ¯ L2_pRn_q for allpb ,ξq PRnˆRn.

Definition 2.5. We define the Wigner transform Wigpf , gqof f and g in L2pRnqby

pWigpf , gqq pb ,ξq “ p2πq´n {2 ż Rn e´i x ¨ξf ´ b`x 2 ¯ g ´ b´x 2 ¯ d x for allpb ,ξq PRnˆRn.

Definition 2.6. We define the Weyl transform W_σf of f in L2pRnqby

pW_σf , gqL2_pRn_q“ p2πq ´n {2 ż Rn ż Rn σpb ,ξq pWigpf , gqq pb ,ξqd b dξ,

for all f and g in L2_p

Rnq.

Definition 2.7. LettG ,˝Gu andtH ,˝Hu be two groups and letϕ : H Ñ Au tpGq. Then GˆH ,˝_ϕ(, where

pg , hq ˝_ϕ`g1, h1˘“`g˝Gϕphqg1, h˝Hh1

˘

,

is a group. This group is called semi-direct product of G and H and it is denoted by G¸_ϕH .

A : X ÑX be a compact operator. Then the operator|A| : X ÑX defined by

|A|“ ?

A˚_A

is positive and compact. So there exists for X an orthonormal basistϕkuconsisting of eigen-vectors of|A|. Let sk be the eigenvalue of|A| : X ÑX corresponding to the eigenvectorϕk.

Then we say that the compact operator A : X ÑX is in the Schatten-von Neumann class Sp, 1ďpă 8if

8

ÿ

k “1

s_kpă 8.

By convention, S8 is taken to be simply the C˚-algebra of all bounded and linear

opera-tor on X . The Schatten-von Neumann class S1is also known as the trace class. Given an

operator A : X ÞÑX in the trace class we can define its trace trpAqas

8

ÿ

k “1

pAϕk,ϕkq_X,

3 Group representation theory

3.1 Localization operators and homogeneous spaces

Let X be a locally compact and Hausdorff topological space and let G be a locally compact and Hausdorff group. We say that G is a left transformation group on X if there exists a continuous mapping GˆX Q pg , xq ÞÑg xPX , such that for all g PG , the mapping X QxÞÑ

g xPX is a homeomorphism of X onto X ,

pg hqx“gph xq, g , hPG , xPX ,

and

e x“x , xPX ,

where e is the identity element in G . The topological space X on which G acts is called a G -space and G is sometimes called a group action on X . Let G be a left transformation group on X such that for all x1and x2in X there exists an element gPG for which x2“g x1. Then

we say that the action of G on X is transitive and we say that X is a homogeneous space. Proposition 3.1. Let X be a homogeneous space on which G acts transitively and let x PX . Then the set Hxdefined by

Hx“tg PG : g x“xu,

is a closed subgroup of G . We call H_x the stability subgroup of G associated to x .

Let G be a locally compact and Hausdorff group and let H be a closed subgroup of G . Let

X “G{H“tg H : g PGu,

and g H , gPG , is the left coset of g in H . Then the action of G on X defined by GˆX Q pg , h Hq ÞÝÑ pg hqH PX , g , hPG ,

3.1 Localization operators and homogeneous spaces

Let G{H be a homogeneous space, where G is a locally compact and Hausdorff group and H is a closed subgroup of G and letµ be a Borel measure on G{H . A Borel measureµ on X

is said to be left quasi-invariant ifµ and µg are equivalent measures on X , where

µgpSq “µpg Sq, gPG ,

for all Borel subsets S of G{H . Notice that we can always equip G{H with a left quasi

invari-ant measure.

We call the canonical surjection of G on G{H the mapping q : GÑG{H defined by G QgÞÑg H PG{H , gPG .

A mappingσ : G{H ÑG such that by qpσpxqq “x , xPG{H ,

is said to be a section on G{H . Assume that G admits a unitary representationπ of G on L2_p Rnq, i .e ., π : GÑU`L2pRnq ˘ , where UpL2_p

Rnqqis the group of all unitary operators on L2_p

Rnq. Suppose that there exists an elementϕPL2pRnqsuch that

ż G {H  pπpσpxqqϕ, fq   2 dµpxq ă 8, @f PL2pRnq.

Then we say thatσ is a strictly admissible section and ϕ is an admissible wavelet.

Remark 3.2. Notice that this definition can be found in[21]. Some authors, for example [33],

talks about square integrable sections. We define the constant c_{σ,H ,ϕ}, by

c_{σ,H ,ϕ}“ ż G {H  pπpσpxqqϕ,ϕq   2 dµpxq.

Let F PL1_{pG{Hq,then we define the linear operator L}

F,σ,H ,ϕ: L2pRnq ÑL2_p Rnqby ` LF,σ,H ,ϕf1, f2 ˘ “ 1 c_{σ,H ,ϕ} ż G {H Fpxq pf₁,πpσpxqqϕq pπpσpxqqϕ, f2qdµpxq, (3.1)

for all f1and f2in L2pRnq.

3.2 Affine group

LF,σ,H ,ϕ: L2pRnq ÑL2_p

Rnqis a bounded linear operator and    L_{F,σ,H ,ϕ}     B pL2_pRn_qqď 1 c_{σ,H ,ϕ}||F ||L1pG {H q.

Proposition 3.4. Let σ be a strictly admissible section. Then the localization operator

LF,σ,H ,ϕ: L2pRnq ÑL2_p

Rnqis in the trace class S₁, and its trace is given by

tr`LF,σ,H ,ϕ˘“ 1

c_{σ,H ,ϕ}

ż

G {H

Fpxqdµpxq.

Remark 3.5. Notice that Proposition 3.3 and Proposition 3.4 are in fact Proposition 25.1,

Proposition 25.3, and Proposition 25.4 on pp.143-145 in[33].

3.2 Affine group

We denote with A the affine group, i .e . A“R¸R`“ tRˆR`,˝Au.

Givenpb , aqandpb1_{, a}1_{qin A, then}

pb , aq ˝A ` b1, a1˘“`b`a b1, a a1˘, and pb , aq´1“ ˆ ´b a, 1 a ˙ .

Proposition 3.6. The left and right Haar measures on A are given by

dµ“d b d a a2 and dν“d b d a a respectively.

3.3 Similitude group

Then, for allpb1_{, a}1_{qin A, we get}

ż A f ``b1, a1˘˝Apb , aq ˘ dµ“ ż R` ż R f `b1`a1b , a1a˘d b d a a2 .

Settingβ“b1_`a1_{b andα“a}1_{a , we have}

ż A f ``b1_{, a}1˘ ˝_Apb , aq˘dµ “ ż R` ż R f `b1`a1b , a1a˘d b d a a2 “ ż R` ż R f pβ,αqdβ d α α2 “ ż A f pb , aqdµ.

Analogously we can prove right invariance for dν.

Remark 3.7. The affine group A is a locally compact non unimodular Hausdorff group.

We can introduce the representation

πApb , aqϕ“T´bD2,aϕ.

Remark 3.8. Notice that we can write

`

W_ϕf˘pb , aq “ pf ,π_Apb , aqϕq_L2_pRn_q

for allpb , aq PA.

Theorem 3.9. π_Ais a unitary irreducible and square-integrable representation of A on the Hardy space H2

`pRq.

3.3 Similitude group

We denote with SIMpnqthe n -dimensonal similitude group, i .e ., SIMpnq “Rn¸ pR`ˆSOpn , Rqq “ tRnˆR`ˆSOpn , Rq,˝SIMpnqu.

Givenpb , a , Rqandpb1_{, a}1_{, R}1_{qin SIM}

pnq, then pb , a , Rq ˝SIMpnq ` b1<sub>, a</sub>1<sub>, R</sub>1˘ “`b`a R b1_{, a a}1_{, R R}1˘ ,

3.3 Similitude group and pb , a , Rq´1“ ´ ´ pa Rq´1b , a´1, R´1 ¯ .

Proposition 3.10. Let d mpRq be the Haar measure on SOpn , Rq, normalized so that

mpSOpn , Rqq “1. Then the left and right Haar measures on SIMpnqare given by dµ“d b d a an `1d mpRq, and dν“d b d a an d mpRq, respectively.

Proof. To prove left invariance, let f be an integrable function on SIMpnqwith respect to the dµ. Then, for allpb1_{, a}1_{, R}1_{qin SIMpnq, we get}

ż SIMpnq f ``b1, a1, R1˘˝SIMpnqpb , a , Rq ˘ dµ “ ż SOpn,Rq ż R` ż Rn f `b1`a1R1b , a1a , R R1˘d b d a an `1d mpRq. Settingβ“b1 `a1<sub>R</sub>1<sub>b ,α</sub> “a1<sub>a and Q</sub> “R1<sub>R , we have</sub> ż SIMpnq f ``b1, a1, R1˘˝SIMpnqpb , a , Rq ˘ dµ “ ż SOpn,Rq ż R` ż Rn f `b1`a1R1b , a1a , R R1˘d b d a an `1d mpRq “ ż SOpn,Rq ż R` ż Rn f pβ,α,Qqdβ dα αn `1d mpQq “ ż SIMpnq f pb , a , Rqdµ.

Analogously we can prove right invariance for dν.

Remark 3.11. The n -dimensional similitude group SIMpnqis a locally compact and non-unimodular Hausdorff group.

We can introduce the representation

3.4 Weyl-Heisenberg group

Remark 3.12. Notice that we can write

`

W_ϕf˘pb , a , Rq “`f ,πSIMpnqpb , a , Rqϕ

˘

L2_pRn_q

for allpb , a , Rq PSIMpnq.

Theorem 3.13. πSIMpnq is a unitary, irreducible and square-integrable representation of

SIMpnqon L2_p

Rnq.

3.4 Weyl-Heisenberg group

We denote by WH the n-dimensional Weyl-Heisenberg group, i .e ., WH“tRnˆRnˆS1,˝_WHu.

Givenpb ,ξ,ϑqandpb1_,ξ1_,ϑ1_{qin WH, then}

pb ,ξ,ϑq ˝WH ` b1<sub>,ξ</sub>1<sub>,ϑ</sub>1˘ “`b`b1<sub>,ξ</sub> `ξ1,ϑ`ϑ1`ξ¨b1˘ , where the third entry in the point is intended modr0, 2πs, and

pb ,ξ,ϑ,a,Rq´1“ p´b ,´ξ,´ϑ`b¨ξq.

Proposition 3.14. The left and right Haar measures on WH are equal and is given by

dµ“d b dξd ϑ.

Proof. To prove left invariance, let f be an integrable function on WH with respect to d µ.

For sake of simplicity, let us assume that f pb ,ξ,¨qis a periodic function with period 2π for fixed but arbitrary b andξ in Rn. Then, for allpb1_,ξ1_,ϑ1_{qin WH, we get}

ż WH f ``b1<sub>,ξ</sub>1<sub>,ϑ</sub>1˘ ˝<sub>WH</sub>pb ,ξ,ϑq˘dµ “ ż S1 ż Rn ż Rn f `b`b1<sub>,ξ</sub> `ξ1,ϑ`ϑ1`ξ¨b1 <sub>mod</sub> r0, 2πs˘d b dξd ϑ. Setting q“b1 `b , p“ξ1`ξ and ϕ“ϑ`ϑ1`ξ¨b1<sub>, we have</sub> ż WH f ``b1,ξ1,ϑ1˘˝WHpb ,ξ,ϑq ˘ dµ “ ż S1 ż Rn ż Rn f `b`b1,ξ`ξ1,ϑ`ϑ1`ξ¨b1 modr0, 2πs˘d b dξd ϑ

3.5 Reduced Heisenberg group “ ż2π 0 ż Rn ż Rn f `b`b1_,ξ `ξ1,ϑ`ϑ1`ξ¨b1˘ d b dξd ϑ “ ż2π`ϑ1_{`ξ¨b</sub>1 ϑ1<sub>`ξ¨b}1 ż Rn ż Rn f pq , p ,ϕqd q d p dϕ “ ż WH f pb ,ξ,ϑqdµ.

With similar computations we can check the right invariance of dµ.

Remark 3.15. The n -dimensional Weyl-Heisenberg group WH is a locally compact,

uni-modular and Hausdorff group. We can introduce the representation

πWHpb ,ξ,ϑqϕ“e

iϑ_T

´bMξϕ.

Remark 3.16. Notice that we can write

`

V_ϕf˘pb ,ξq “ p2πq´n {2e´i b ¨ξpf ,π_WHpb ,ξ,0qϕqL2_pRn_q

for allpb ,ξ,0q PWH.

Theorem 3.17. π_WHis a unitary, irreducible and square-integrable representation of WH on L2pRnq.

3.5 Reduced Heisenberg group

We denote by H the n-dimensional reduced Heisenberg group, i .e . H“tRnˆRnˆR,˝Hu.

Givenpb ,ξ,ϑqandpb1_,ξ1_,ϑ1_{qin H, then}

pb ,ξ,ϑq ˝_H`b1<sub>,ξ</sub>1<sub>,ϑ</sub>1˘ “`b`b1<sub>,ξ</sub> `ξ1,ϑ`ϑ1`ξ¨b1˘ , and pb ,ξ,ϑq´1“ p´b ,´ξ,´ϑ`b¨ξq.

Proposition 3.18. The left and right Haar measures on H are equal and is given by

3.5 Reduced Heisenberg group

Proof. To prove left invariance, let f be an integrable function on H with respect to the d µ.

Then, for allpb1_,ξ1_,ϑ1_{qin H, we get}

ż H f ``b1,ξ1,ϑ1˘˝<sub>H</sub>pb ,ξ,ϑq˘dµ “ ż R ż Rn ż Rn f `b`b1<sub>,ξ</sub> `ξ1,ϑ`ϑ1`ξ¨b1˘ d b dξd ϑ. Setting q“b1 `b , p“ξ1`ξ and ϕ“ϑ`ϑ1`ξ¨b1<sub>, we have</sub> ż H f ``b1,ξ1,ϑ1˘˝Hpb ,ξ,ϑq ˘ dµ “ ż R ż Rn ż Rn f `b`b1,ξ`ξ1,ϑ`ϑ1`ξ¨b1˘d b dξd ϑ “ ż R ż Rn ż Rn f pq , p ,ϕqd q d p dϕ “ ż H f pb ,ξ,ϑqdµ.

With similar computations we can check the right invariance of dµ.

Remark 3.19. The n -dimensional Weyl-Heisenberg group H is a locally compact,

unimod-ular and Hausdorff group.

We can introduce the representation

πHpb ,ξ,ϑqϕ“e

iϑ_T

´bMξϕ.

Remark 3.20. Notice that we can write

`

V_ϕf˘pb ,ξq “ p2πq´n {2e´i b ¨ξ

pf ,π_Hpb ,ξ,0qϕq_L2_pRn_q

for allpb ,ξ,0q PH.

Theorem 3.21. π_H is a unitary, irreducible and non-square-integrable representation of H on L2_p

Rnq.

Theorem 3.22. Let H be the closed subgroup of H defined as

H“ tp0, 0,ϑq PHu.

Letσ : H{HÑH be the Borel section given by

3.6 Affine Heisenberg group

Thenσ is a strictly admissible section. Furthermore, every non zero ϕPL2_p

Rnqis an admis-sible wavelet for this strictly admisadmis-sible section, i .e . givenϕPL2pRnqsuch that

   ϕ    “1, we have c_{ϕ,H ,σ}“ ż Rn ż Rn  pϕ,π_Hpσpb ,ξqqϕq   2 d b dξ“ p2πqn.

Remark 3.23. Notice that we can write

`

V_ϕf˘pb ,ξq “ p2πq´n {2e´i b ¨ξ

pf ,π_Hpσpb ,ξqqϕq_L2_pRn_q

for allpb ,ξq PH{H .

3.6 Affine Heisenberg group

We denote by AH the n-dimensonal affine Heisenberg group, i .e . AH“H¸ pR`ˆSOpn , Rqq “ tRnˆRnˆRˆR`ˆSOpn , Rq,˝AHu.

Givenpb ,ξ,ϑ,a,Rqandpb1_,ξ1_,ϑ1_{, a}1_{, R}1_{qin AH, then}

pb ,ξ,ϑ,a,Rq ˝AH ` b1<sub>,ξ</sub>1<sub>,ϑ</sub>1<sub>, a</sub>1<sub>, R</sub>1˘ “`b`a R b1,ξ`a´1Rξ1,ϑ`ϑ1`ξ¨`a R b1˘, a a1, R R1˘, and pb ,ξ,ϑ,a,Rq´1“`´a´1R´1b ,´a R´1ξ,´ϑ`b¨ξ,a´1, R´1˘.

Proposition 3.24. Let d mpRq be the Haar measure on SOpn , Rq, normalized so that

mpSOpn , Rqq “1. Then the left and right Haar measures on AH are equal and is given by

dµ“d b dξd ϑd a

a d mpRq.

Proof. To prove left invariance, let f be an integrable function on AH with respect to the dµ. Then, for allpb1_,ξ1_,ϑ1_{, a}1_{, R}1_{qin AH, we get}

ż AH f ``b1,ξ1,ϑ1, a1, R1˘˝AHpb ,ξ,ϑ,a,Rq ˘ dµ “ ż SOpn,Rq ż R` ż R ż Rn ż Rn f `b1`a1R1b ,ξ1`a1´1R1ξ,ϑ1`ϑ`ξ1¨`a1R1b˘, a1a , R1R˘

3.6 Affine Heisenberg group ˆd b dξd ϑd a a d mpRq. Setting q “b1_{`a</sub>1<sub>R</sub>1<sub>b , p “ξ</sub>1<sub>`a}1´1_R1_{ξ, ϕ“ϑ}1_`ϑ`ξ1_{¨ pa}1_R1_bq,α“a1_{a and Q“R}1_{R ,} we have ż AH f ``b1,ξ1,ϑ1, a1, R1˘˝AHpb ,ξ,ϑ,a,Rq ˘ dµ “ ż SOpn,Rq ż R` ż R ż Rn ż Rn f `b1`a1R1b ,ξ1`a1´1R1ξ,ϑ1`ϑ`ξ1¨`a1R1b˘, a1a , R1R˘ ˆd b dξd ϑd a a d mpRq “ ż SOpn,Rq ż R` ż R ż Rn ż Rn f pq , p ,ϕ,α,Qqd q d p dϕdα α d mpQq “ ż AH f pb ,ξ,ϑ,a,Rqdµ.

Analogously we can prove the right invariance of dµ.

Remark 3.25. The n -dimensional affine Heisenberg AH group is a locally compact,

uni-modular and Hausdorff group. We can introduce the representation

πAHpb ,ξ,ϑ,a,Rqϕ“e

iϑ_T

´bMξD2,a Rϕ. (3.2)

Theorem 3.26. π_AHis a unitary, irreducible and non-square-integrable representation of AH on L2pRnq.

Theorem 3.27. Let H “ tp0, 0,ϑ,a,Rq PAHube a closed subgroup of AH, let a : Rn QξÞÑ

apξq PR`and R : RnQξÑRpξq PSOpn , Rqbe two piecewise differentiable functions such

that we can find a fixedp1, 2q-tensor F and a fixedp1, 1q-tensor G such that

´ apξq´1Rpξq ¯i j “F i j lξ l `G_ji,

and letη_ζ: RnÑRn be the function given by

ηζpξq “apξqRpξq´1pζ´ξq,

is such thatη_ζpRnq “Rn, for allζPRn. Letσ : AH{H ÑAH be the section given by

3.6 Affine Heisenberg group and letϕPL2_p Rnqbe such that c_ϕ“ ż Rn  ϕ_ppηq   2 dη   det ´ F_{j k}i ηj _`δi k ¯   ă 8.

Then we have a resolution of the identity formula, i .e ., c_ϕpf , gqL2_pRn_q“ ż Rn ż Rn pf ,π_AHpσpb ,ξqqqL2_pRn_qpπ_AHpσpb ,ξqq, gq_L2_pRn_qd b dξ.

4 Stockwell transforms

Definition 4.1. Let 1ďs ă 8and A : Rn _{QξÞÑApξq “A}

ξPGLpn , Rq. Then we define the Stockwell transform S_{s ,A,ϕ}f of the signal f with respect to the windowϕ by

` Ss ,A,ϕf˘pb ,ξq “ p2πq´n {2det A_ξ   ´1{sż Rn f pxqe´i x ¨ξ_ϕ´_A´1 ξ px´bq ¯ d x for allpb ,ξq PRnˆRn.

Remark 4.2. Notice that we can write

` Ss ,A,ϕf ˘ pb ,ξq “det A<sub>ξ</sub>   1{2´1{s` S2,A,ϕf ˘ pb ,ξq.

Remark 4.3. Choosing n “ 1, s “ 1 and A : R Q ξ ÞÑ 1{ξ P Rz t0u, we recover the 1-dimensional Stockwell transform defined in 1.3.

We can elucidate the link between the analyzed signal f and the analyzing windowϕ. Proposition 4.4. Let f andϕ be signals in L2pRnq. Then we have

` Ss ,A,ϕf ˘ pb ,ξq “e´i b ¨ξ`Ss ,A, f ϕ ˘´ ´A´1_ξ b ,´At_ξξ ¯ , b ,ξPRn.

Proof. It is sufficient to prove the proposition for s“2. In this case we can write

` S2,A,ϕf˘pb ,ξq “ p2πq´n {2 ´ f , M<sub>ξ</sub>T<sub>´b</sub>D2,A<sub>ξ</sub>ϕ ¯ L2<sub>pR</sub>n<sub>q</sub> “ p2πq´n {2 ´ D<sub>2,A</sub>´1 ξ TbM´ξf ,ϕ ¯ L2<sub>pR</sub>n<sub>q</sub> “ p2πq´n {2 ´ T<sub>A</sub>´1 ξ bM´A t ξξD2,A´1ξ f ,ϕ ¯ L2<sub>pR</sub>n<sub>q</sub> “ p2πq´n {2e´i b ¨ξ´<sub>M</sub> ´A<sub>ξ</sub>tξTA´1<sub>ξ</sub> bD2,A´1<sub>ξ</sub> f ,ϕ ¯ L2<sub>pR</sub>n<sub>q</sub> “e´i b ¨ξ p2πq´n {2 ´ ϕ,M´At<sub>ξ</sub>ξTA´1<sub>ξ</sub> bD2,A´1<sub>ξ</sub> f ¯ L2<sub>pR</sub>n<sub>q</sub> “e´i b ¨ξ`S2,A, f ϕ ˘´ ´A´1_ξ b ,´A_ξtξ ¯ .

Stockwell transforms

We give here an alternative formulation of Stockwell transforms. Proposition 4.5. Let f be a signal in L2_p

Rnqand letϕ be a window in L2_p

Rnq. Then ` Fb ÞÑζSs ,A,ϕf ˘ pζ,ξq “`T_ξF f˘pζq ˆ D s s ´1,pA´1ξ q tFϕ ˙ pζq, ζ,ξPRn, which implies ` S_{s ,A,ϕ}f˘pb ,ξq “ ´ M´ξ ´ f ˚M_ξDs ,´Aξϕ ¯¯ pbq, b ,ξPRn.

Proof. Thanks to the Fubini-Tonelli theorem and to the change of variable y “A´1_ξ px´bq, we get ` Fb ÞÑζSs ,A,ϕf˘pζ,ξq “ p2πq´n {2 ż Rn e´i b ¨ζ`S_{s ,A,ϕ}f˘pb ,ξqd b “ p2πq´n {2 ż Rn e´i b ¨ζ p2πq´n {2det Aξ   ´1{s ˆ ż Rn f pxqe´i x ¨ξϕ ´ A´1_ξ px´bq ¯ d x d b “ p2πq´ndet Aξ   ´1{sż Rn e´i x ¨ξ_f pxq ż Rn e´i b ¨ζ_ϕ´_A´1 ξ px´bq ¯ d b d x “ p2πq´ndet Aξ   1´1{sż Rn e´i x ¨ξ_f pxq ż Rn e´ipx ´Aξyq¨ζ_ϕ pyqd y d x “ p2πq´ndet A_ξ 1´1{sż Rn e´i x ¨ξ_f pxqe´i x ¨ζ ż Rn ei y ¨Atξζϕ_{pyqd y d x} “ p2πq´n {2det A_ξ   1´1{sż Rn e´i x ¨ξf pxqe´i x ¨ζϕp ´ A_ξtζ ¯ d x “ p2πq´n {2det A_ξ 1´1{s p ϕ´At_ξζ ¯ż Rn e´i x ¨pζ`ξq<sub>f</sub> pxqd x “det A<sub>ξ</sub>   1´1{s p f pζ`ξqϕp ´ At_ξζ ¯ “`T<sub>ξ</sub>F f˘pζq ˆ D s s ´1,pA ´1 ξ q tF ˙ ϕpζq “`F M´ξf ˘ pζq ´ F Ds ,A_ξϕ ¯ pζq “`F M_´ξf˘pζq ´ F Ds ,´A_ξϕ ¯ pζq, ζ,ξPRn. The second formula follows because

` Fb ÞÑζSs ,A,ϕf ˘ pζ,ξq “`F M´ξf ˘ pζq ´ F Ds ,´A_ξϕ ¯ pζq

Stockwell transforms “ ´ F ´ M´ξf ˚Ds ,´A_ξϕ ¯¯ pζq, ζ,ξPRn. And via the Fourier inversion formula, we get

` Ss ,A,ϕf ˘ pb ,ξq “ ´ M´ξf ˚Ds ,´A_ξϕ ¯ pbq “ ´ M_´ξ ´ f ˚M_ξDs ,´A_ξϕ ¯¯ pbq, b ,ξPRn.

Notice that the Stockwell transform describes the behaviour of the signal f under convolu-tion with a dilated and modulated windowϕ. In fact,

  ` Ss ,A,ϕf ˘ pb ,ξq“    ´ f ˚M_ξDs ,´A_ξϕ ¯ pbq    , b ,ξPR n_.

With the new formulation of the Stockwell transforms in place, we can give the following estimate. Proposition 4.6. Letϕ be in Lp_p Rnq,1ďpď 8. Then    ϕs ,b ,ξ     Lp_pRn_q“  det Aξ   1{p ´1{s   ϕ     Lp_pRn_q, where ϕs ,b ,ξ“M_ξT´bDs ,Aϕ, and   ` Ss ,A,ϕf˘pb ,ξqď p2πq´n {2  det A_ξ 1{p ´1{s   f     Lp 1_pRn_q    ϕ     Lp_pRn_q, b ,ξPR n_,

where p1_{is the conjugated index of p.}

Proof. Let 1ďp ă 8. Then, thanks to the Fubini-Tonelli theorem and to the change of variable y“A´1_ξ px´bq, we get    ϕs ,b ,ξ     p Lp_pRn_q“ ż Rn  ϕs ,b ,ξpxq   p d x “ ż Rn  M_ξT_´bD_{s ,A}ϕpxq   p d x “ ż Rn   e i x ¨ξ det A_ξ   ´1{s ϕ´A´1_ξ px´bq ¯   p d x “ ż Rn  det A_ξ   ´p {s ϕ ´ A´1_ξ px´bq ¯   p d x

Stockwell transforms “ ż Rn  det A_ξ 1´p {s ϕpyq   p d y “det Aξ   1´p {s   ϕ     p Lp_pRn_q. If p“ 8, we get    ϕs ,b ,ξ     L8_pRn_q“sup x PRn  ϕs ,b ,ξpxq   “sup x PRn  M_ξT_´bD_{s ,A}ϕpxq   “sup x PRn   e i x ¨ξ det A_ξ   ´1{s ϕ´A´1_ξ px´bq ¯   “sup x PRn     det A_ξ   ´1{s ϕ´A´1_ξ px´bq ¯   “det A_ξ   ´1{s sup x PRn   ϕ ´ A´1_ξ px´bq ¯   “det A_ξ   ´1{s   ϕ     L8_pRn_q.

Let 1ďpď 8. Then applying Holder’s inequality, we have   ` Ss ,A,ϕf˘pb ,ξq“     p2πq´n {2det A_ξ ´1{sż Rn f pxqe´i x ¨ξ_ϕ´_A´1 ξ px´bq ¯ d x     ď p2πq´n {2det A_ξ   ´1{sż Rn     f pxqe´i x ¨ξ_ϕ´_A´1 ξ px´bq ¯   d x “ p2πq´n {2 ż Rn  f pxq       ei x ¨ξ det A_ξ   ´1{s ϕ´A´1_ξ px´bq ¯   d x ď p2πq´n {2  f     Lp 1_pRn_q    ϕs ,b ,_ξ     Lp_pRn_q “ p2πq´n {2det A_ξ   1{p ´1{s   f     Lp 1_pRn_q    ϕ     Lp_pRn_q.

We remark that if s“p , then Proposition 4.6 gives a weighted L8_{-estimate of the Stockwell}

transform. In fact, sup pb ,ξqPRn_ˆRn    ` Ss ,A,ϕf ˘ pb ,ξqdet A_ξ   1 s´ 1 p   ď p2πq ´n {2   f     Lp 1_pRn_q    ϕ     Lp_pRn_q.

4.1 Stockwell transforms and affine Heisenberg group

4.1 Stockwell transforms and affine Heisenberg group

Proposition 4.7. Let a : Rn ÑR`and R : Rn ÑSOpn , Rqbe two piecewise differentiable

functions. Then we can introduce A : RnÑGLpn , Rqas Apξq “apξqRpξq, and write ` S_{s ,A,ϕ}f˘pb ,ξq “ p2πq´n {2apξq   n {2´n{s e´i b ¨ξ pf ,π_AHpσpb ,ξqqϕq,

whereπ_AHis defined in (3.2), Theorem 3.26 and Theorem 3.27.

Proof. Via direct computations, we get

` Ss ,A,ϕf ˘ pb ,ξq “det A<sub>ξ</sub>   1{2´1{s S2,A,ϕf pb ,ξq “detpapξqRpξqq   1{2´1{s S2,a R ,ϕf pb ,ξq “ p2πq´n {2apξq   n {2´n{s` f , M_ξT_´bD2,a pξqRpξqϕ ˘ “ p2πq´n {2apξq   n {2´n{s e´i b ¨ξ`f , T´bMξD2,a pξqRpξqϕ ˘ “ p2πq´n {2apξq   n {2´n{s e´i b ¨ξ pf ,π_AHpσpb ,ξqqϕq.

4.2 Gabor transforms

We recall here Definition 2.2, i .e ., we define the Gabor transform V_ϕf of the signal f with

respect to the windowϕ by

` V<sub>ϕ</sub>f˘pb ,ξq “ p2πq´n {2`f , M_ξT´bϕ ˘ L2_pRn_q “ p2πq´n {2 ż Rn e´i x ¨ξ_f pxqϕppx´bqqd x for allpb ,ξq PRnˆRn. Proposition 4.8. We can write

` Ss ,A,ϕf ˘ pb ,ξq “ ´ VD_{s ,Aξ}ϕf ¯ pb ,ξq for allpb ,ξq PRnˆRn.

4.2 Gabor transforms

Proof. Via direct computation, we get

` Ss ,A,ϕf ˘ pb ,ξq “ p2πq´n {2 ´ f , M_ξT´bDs ,A_ξϕ ¯ L2_pRn_q“ ´ VD_{s ,Aξ}ϕf ¯ pb ,ξq.

Proposition 4.9. Let f be a signal in L2pRnqand letϕ be a window in L2pRnq. Then we can

write ` Ss ,A,ϕf ˘ pb ,ξq “ ˆ D1_{s ,A} ξD 2 s ,pA´1_ξ qt ´ V_ϕ ´ D_{s ,A}´1 ξ f ¯¯˙ pb ,ξq “ ´ V_ϕ ´ D_{s ,A}´1 ξ f ¯¯ ´ A´1_ξ b , A_ξtξ ¯ , b ,ξPRn.

Proof. Using Proposition 4.8, we get

` Ss ,A,ϕf ˘ pb ,ξq “ ´ VD_{s ,Aξ}ϕf ¯ pb ,ξq “ ´ f , M_ξT´bDs ,A_ξϕ ¯ L2_pRn_q “ ´ f , M_ξDs ,A_ξT´A´1_ξ bϕ ¯ L2_pRn_q “ ´ f , D_{s ,Aξ}MAt ξξT´A´1_ξ bϕ ¯ L2_pRn_q “ ´ D_{s ,A}´1 ξ f , MA t ξξT´A´1_ξ bϕ ¯ L2_pRn_q “ ´ V_ϕ ´ D_{s ,A}´1 ξ f ¯¯ ´ A´1_ξ b , A_ξtξ ¯ “ ˆ D1_{s ,A} ξD 2 s ,pA´1_ξ qt ´ V_ϕ ´ D_{s ,A}´1 ξ f ¯¯˙ pb ,ξq, b ,ξPRn.

Remark 4.10. Notice that, thanks to Proposition 4.9 when A : Rn _{Q ξÞÑ Apξq “ A} ξ P SOpn , Rq ĂGLpn , Rq, we get ` S_{s ,A,ϕ}f˘pb ,ξq “ ´ Ds ,Aξ ´ V_ϕ ´ D_{s ,A}´1 ξ f ¯¯¯ pb ,ξq “ ´ V_ϕ ´ D_{s ,A}´1 ξ f ¯¯ ´ A´1_ξ b , A´1_ξ ξ ¯ , b ,ξPRn.

4.3 Moritoh wavelet transforms

4.3 Moritoh wavelet transforms

We recall here Definition 2.4, i .e . let R : Rn QξÞÑRpξq “R_ξPSOpn , Rq. Then we define the Moritoh wavelet W1

|ξ|R´1,ϕf of the signal f with respect to the windowϕ by

´ W1 |ξ|R´1,ϕf ¯ pb ,ξq “|ξ|n {2 ż Rn f pxqϕ`|ξ| Rξpx´bq˘d x “ ´ f , T´bD_2,1 |ξ|R ´1 ξ ϕ ¯ L2_pRn_q for allpb ,ξq PRnˆRn.

Notice that, setting A_ξ“_|ξ|1R_ξ´1, we can write

´ WA_ξ,ϕf ¯ pb ,ξq “ ´ f , T_´bD2,A_ξϕ ¯ L2_pRn_q.

Proposition 4.11. Let 1ďsă 8, let R : RnQξÞÑRpξq “R_ξPSOpn , Rq, and let A_ξ“_|ξ|1R´1

ξ . Then we can write

` Ss ,A,ϕf ˘ pb ,ξq “det A_ξ   1{2´1{s p2πq´n {2e´i b ¨ξ ˆ WA_ξ,M pA´1_{ξ q}tξϕf ˙ pb ,ξq for allpb ,ξq PRnˆRn.

Proof. First we can show that

` S2,A,ϕf ˘ pb ,ξq “ p2πq´n {2e´i b ¨ξ ˆ WA_ξ,M pA´1_{ξ q}tξϕf ˙ pb ,ξq, b ,ξPRn.

In fact, via direct computation, we get for all b andξ in Rn,

p2πq´n {2e´i b ¨ξ ˆ WAξ,M pA´1_{ξ q}tξϕf ˙ pb ,ξq “ p2πq´n {2e´i b ¨ξ ˆ f , T´bD2,A_ξM_pA´1 ξ q t ξϕ ˙ “ p2πq´n {2e´i b ¨ξ ˆ f , T´bM_At ξpA´1ξ q t ξD2,A_ξϕ ˙ “ p2πq´n {2e´i b ¨ξ´_{f , T} ´bMξD2,Aξϕ ¯ “ p2πq´n {2e´i b ¨ξ ´ f , e´i b ¨ξM_ξT´bD2,A_ξϕ ¯ “ p2πq´n {2e´i b ¨ξ_ei b ¨ξ´ f , M_ξT_´bD2,A_ξϕ ¯ “`S2,A,ϕf˘pb ,ξq.

4.3 Moritoh wavelet transforms

Thanks to Remark 4.2, we get

` Ss ,A,ϕf ˘ pb ,ξq “det A<sub>ξ</sub>   1{2´1{s` S_2,A,ϕf˘pb ,ξq “det A_ξ   1{2´1{s p2πq´n {2e´i b ¨ξ ˆ WA_ξ,M pA´1_{ξ q}tξϕf ˙ pb ,ξq, b ,ξPRn.

5 Continuous inversion formulas

Proposition 5.1. Let f be a signal in L2pRnqand letϕPL1pRnq XL2pRnqbe such that

ż Rn ϕpxqd x“1. Then ż Rn ` S_{s ,A,ϕ}f˘pb ,ξqd b “det Aξ   1´1{s p f pξq, ξPRn.

Proof. Thanks to the Fubini theorem and the change of variable y “A_ξpx´bqwe get

ż Rn ` Ss ,A,ϕf ˘ pb ,ξqd b “ ż Rn ˆ p2πq´n {2det A_ξ   ´1{sż Rn f pxqe´i x ¨ξϕ ´ A´1_ξ px´bq ¯ d x ˙ d b “ p2πq´n {2 ż Rn ż Rn f pxqe´i x ¨ξ_ϕ´_A´1 ξ px´bq ¯ det Aξ   ´1{s d x d b “ p2πq´n {2 ż Rn e´i x ¨ξf pxqdet A_ξ   ´1{sˆż Rn ϕ´A´1_ξ px´bq ¯ d b ˙ d x “ p2πq´n {2 ż Rn e´i x ¨ξf pxqdet A_ξ   1´1{sˆż Rn ϕpyqd y ˙ d x “det A_ξ   1´1{s p f pξq, ξPRn. Proposition 5.2. Let f ,ϕPL2_p

Rnqand A : RnQξÞÑAξPGLpn , Rq. Then we have

` S_{s ,A,ϕ}f˘pb ,ξq “det Aξ   1´1{s e´i b ¨ξ´_F´1 ζÞÑb fξ,Aξ ¯ pbq, b ,ξPRn, where f_ξ,Aξpζq “ pfpζqϕp ´ A_ξtpζ´ξq ¯ .

Continuous inversion formulas

Proof. First notice that for all b andξ in Rn_,

F T´bMξD2,A_ξϕ“M´bT´ξD_2,pA´1

ξ q tFϕ,

So, by the Plancherel formula,

´ f , T_´bM_ξD2,A_ξϕ ¯ L2_pRn_q “ ´ F f , F T´bMξD2,A_ξϕ ¯ L2_pRn_q “ ˆ F f , M´bT´ξD_2,pA´1 ξ q tFϕ ˙ L2_pRn_q “det Aξ   1{2ż Rn p f pζqei b ¨ζϕp ´ At_ξpζ´ξq ¯ dζ “ p2πqn {2det A_ξ   1{2 p2πq´n {2 ż Rn p f pζqei b ¨ζϕp ´ A_ξtpζ´ξq ¯ dζ “ p2πqn {2det Aξ   1{2´ F´1_ζÞÑbf_ξ,Aξ ¯ pbq, where f_ξ,Aξpζq “ pfpζqϕp ´ A_ξtpζ´ξq ¯ . Since we can write

` S2,A,ϕf ˘ pb ,ξq “ p2πq´n {2e´i b ¨ξ ´ f , T´bMξD2,A_ξϕ ¯ L2_pRn_q, b ,ξPR n_,

thanks to Remark 4.2, we get

` Ss ,A,ϕf ˘ pb ,ξq “det A<sub>ξ</sub>   1{2´1{s` S2,A,ϕf ˘ pb ,ξq “det A_ξ   1{2´1{s p2πq´n {2e´i b ¨ξ ´ f , T´bMξD2,Aξϕ ¯ L2_pRn_q “det A_ξ   1{2´1{s p2πq´n {2e´i b ¨ξ p2πqn {2det A_ξ   1{2 F´1_ζÞÑb f_ξ,Aξpbq “det A_ξ   1´1{s e´i b ¨ξ´_F´1 ζÞÑb fξ,Aξ ¯ pbq, b ,ξPRn.

5.1 Constant matrices

5.1 Constant matrices

In this subsection we want to study matrix-valued maps A : RnQξÞÑA_ξPGLpn , Rqsuch that

A_ξ“APGLpn , Rq @ξPRn, (5.1)

and their associated Stockwell transforms.

Lemma 5.3. Let A : RnQξÞÑA_ξPGLpn , Rqbe such that A_ξ“APGLpn , Rq, @ξPRn,

and letϕPL2pRnq. Then

ż Rn  ϕ_p ` Atpζ´ξq˘ 2 |det A| d ξ“  ϕ     2 L2_pRn_q.

Proof. Letη“Atpζ´ξq. Then J_ηpξq “ ´At,

so

dη“det J_ηpξq 

dξ“|det A| d ξ. Now we can write

ż Rn  ϕ_p ` Atpζ´ξq˘ 2 |det A| d ξ“ ż Rn  ϕ_ppηq   2 dη“  ϕ     2 L2_pRn_q.

Proposition 5.4. Let 1ďsă 8, let A : RnQξÞÑA_ξPGLpn , Rqbe such that A_ξ“APGLpn , Rq, @ξPRn,

and letϕPL2pRnq. Then

   ϕ     2 L2_pRn_qpf , gqL2_pRn_q “ ż Rn ż Rn ` Ss ,A,ϕf ˘ pb ,ξq`Ss ,A,ϕg ˘ pb ,ξqd b |det A|2{s ´1dξ.

5.2 Diagonal matrices 5.3, we get ż Rn ż Rn ` Ss ,A,ϕf ˘ pb ,ξq`Ss ,A,ϕg ˘ pb ,ξqd b |det A|2{s ´1dξ “ ż Rn ż Rn |det A|1´1{se´i b ¨ξ ´ F´1_ζÞÑb f_ξ,A ¯ pbq ˆ|det A|1´1{se´i b ¨ξ_F´1 ζÞÑbgξ,Apbqd b |det A|2{s ´1dξ “ ż Rn ż Rn ´ F´1_ζÞÑbf_ξ,A ¯ pbq ´ F´1_ζÞÑbg_ξ,A ¯ pbqd b |det A| d ξ “ ż Rn ż Rn

f_ξ,Apζqg_ξ,Apζqdζ |det A|d ξ

“ ż Rn ż Rn p fpζqϕppAtpζ´ξqqpgpζqϕppAtpζ´ξqqdζ |det A|d ξ “ ż Rn p fpζqgppζq ż Rn  ϕ_p ` Atpζ´ξq˘ 2 |det A| d ξ d ζ “  ϕ     2 L2_pRn_q ż Rn p fpζqgppζqdζ “  ϕ     2 L2_pRn_qpf , gqL2_pRn_q.

5.2 Diagonal matrices

In this subsection we want to study matrix-valued maps A : RnQξÞÑA_ξPGLpn , Rqin the form ˆ ´ An ˆn_ξ ¯´1˙t “ ¨ ˚ ˚ ˚ ˝ ξ1 ξ2 . . . ξn ˛ ‹ ‹ ‹ ‚ , ξ1, . . . ,ξn‰0, @ξPRn, (5.2)

and their associated Stockwell transforms. For the sake of clarity, we write A_ξ instead of

An ˆn_ξ .

Lemma 5.5. Let A : RnQξÞÑA_ξPGLpn , Rqbe such that

At_ξ“ ¨ ˚ ˚ ˚ ˝ 1 ξ1 1 ξ2 . . . 1 ξn ˛ ‹ ‹ ‹ ‚ , ξ1, . . . ,ξn‰0, @ξPRn,

5.2 Diagonal matrices and letϕPL2_p Rnq. Then ż Rn   ϕp ´ At_ξpζ´ξq ¯   2 det A_ξdξ“ ż Rn  ϕ_ppη´1q   2 det A_ηdη, where 1“ p1, 1, . . . , 1q PRn.

Proof. Notice that

At_ξpζ´ξq “At_ξζ´1,

where 1“ p1, 1, . . . , 1q PRn, and letη“At_ξζ, then

J_ηpξq “ ¨ ˚ ˚ ˚ ˚ ˝ ´ζ_ξ12 1 ´ζ_ξ22 2 . . . ´ζn ξ2 n ˛ ‹ ‹ ‹ ‹ ‚ “ ¨ ˚ ˚ ˚ ˚ ˝ ´η1ξ1 ξ2 1 ´η_ξ2ξ22 2 . . . ´ηn_ξξ2n n ˛ ‹ ‹ ‹ ‹ ‚ “ ¨ ˚ ˚ ˚ ˝ ´η_ξ1 1 ´η_ξ2 2 . . . ´ηn ξn ˛ ‹ ‹ ‹ ‚ , so dη“det J_ηpξq  dξ“ n ź j “1  ηj    ξj   dξ.

Now we can write

ż Rn   ϕp ´ At_ξpζ´ξq ¯   2 _dξ śn j “1  ξj   “ ż Rn  ϕ_ppη´1q   2 dη śn j “1  ηj   .

Proposition 5.6. Let 1ďsă 8, let A : RnQξÞÑA_ξPGLpn , Rqbe such that

At_ξ“ ¨ ˚ ˚ ˚ ˝ 1 ξ1 1 ξ2 . . . 1 ξn ˛ ‹ ‹ ‹ ‚ , ξ1, . . . ,ξn‰0, @ξPRn,