Segal-Bargmann transform and Paley-Wiener theorems on M(2)

Segal–Bargmann transform and Paley–Wiener theorems on M(2) M(2) M(2)

E K NARAYANAN and SUPARNA SEN

Department of Mathematics, Indian Institute of Science, Bangalore 560 012, India E-mail: naru@math.iisc.ernet.in; suparna@math.iisc.ernet.in

MS received 28 July 2009; revised 9 November 2009

Abstract. We study the Segal–Bargmann transform onM(2).The range of this trans- form is characterized as a weighted Bergman space. In a similar fashion Poisson integrals are investigated. Using a Gutzmer’s type formula we characterize the range as a class of functions extending holomorphically to an appropriate domain in the complexification ofM(2).We also prove a Paley–Wiener theorem for the inverse Fourier transform.

Keywords. Segal–Bargmann transform; Poisson integrals; Paley–Wiener theorem.

1. Introduction

Consider the following results from Euclidean Fourier analysis:

(I) A functionf ∈L²(Rⁿ)admits a factorizationf (x)=g∗p_t(x)whereg∈L²(Rⁿ) andp_t(x)= ¹

(4πt)ⁿ2e^−|x|

2

4t (the heat kernel onRⁿ) if and only iff extends as an entire function toCⁿand we have_(2πt)¹n/2

Cⁿ|f (z)|²e^−|y|

2

2t dxdy <∞(z=x+iy).In this case we also have

g²₂= 1 (2πt)^n/2

Cⁿ|f (z)|²e^−|y|

2 2t dxdy.

The mapping g→g∗pt is called the Segal–Bargmann transform and the above says that the Segal–Bargmann transform is an unitary map from L²(Rⁿ) ontoO(Cⁿ) L²(Cⁿ, μ),where dμ(z)= _(2πt)¹n/2e^−|y|

2

2t dxdyandO(Cⁿ)denotes the space of entire functions onCⁿ.

(II) A functionf ∈L²(R)admits a holomorphic extension to the strip{x+iy:|y|< t} such that

sup

|y|≤s

R|f (x+iy)|²dx <∞ ∀s < t if and only if

e^s|ξ|f (ξ)˜ ∈L²(R) ∀s < t wheref˜denotes the Fourier transform off.

169

(III) Anf ∈L²(Rⁿ)admits an entire extension toCⁿsuch that

|f (z)| ≤C_N(1+ |z|)^−Ne^R|Imz|∀z∈Cⁿ

if and only iff˜∈Cc^∞(Rⁿ)and suppf˜⊆B(0, R),whereB(0, R)is the ball of radiusR centered around 0 inRⁿ.

In this paper we aim to prove similar results for the non-commutative groupM(2)= R²SO(2).Some remarks are in order.

As noted above the mapg→g∗p_t in (I) is called the Segal–Bargmann transform.

This transform has attracted a lot of attention in recent years mainly due to the work of Hall [3] where a similar result was established for an arbitrary compact Lie groupK.Let q_t be the heat kernel onK and letK_C be the complexification ofK.Then Hall’s result (Theorem 2 in [3]) states that the mapf → f ∗q_t is an unitary map fromL²(K)onto the Hilbert space ofν-square integrable holomorphic functions onK_Cfor an appropriate positiveK-invariant measureνonK_C.Soon after Hall’s paper a similar result was proved for compact symmetric spaces by Stenzel in [9]. We also refer to [4–7] for similar results for other groups and spaces.

The second result (II) is originally due to Paley and Wiener. Let

Ht = {f ∈L²(R), f has a holomorphic extension to|Imz|< tand sup

|y|≤s

R|f (x+iy)|²dx <∞ ∀s < t}. Then

t>0Ht may be viewed as the space of all analytic vectors for the regular repre- sentation ofRonL²(R).This point of view was further developed by Goodman (see [1]

and [2]) who studied analytic vectors for representations of Lie groups. The theorem of Paley–Wiener (II) characterizes analytic vectors for the regular representation ofRvia a condition on the Fourier transform.

The third result (III) is the classical Paley–Wiener theorem. For a long time the Paley–

Wiener theorem has been looked at as a characterization of the image (under Fourier transform) ofC_c^∞functions on the space you are interested in. Recently, using Gutzmer’s formula, Thangavelu [11] has proved a Paley–Wiener type result for the inverse Fourier transform (see [8] also for a similar result).

The plan of this paper is as follows: In the remaining of this section we recall the representation theory and Plancherel theorem ofM(2)and we prove the unitarity of the Segal–Bargmann transform. In the next section, we study generalized Segal–Bargmann transform and prove an analogue of Theorems 8 and 10 in [3]. Section 3 is devoted to a study of Poisson integrals onM(2).This section is modeled after the work of Goodman [1] and [2]. In the final section, we establish a Paley–Wiener type result for the inverse Fourier transform onM(2).

The rigid motion groupM(2)is the semi-direct product ofR²withSO(2)(which will be identified with the circle groupS¹) with the group law

(x,e^iα)·(y,e^iβ)=(x+e^iαy,e^i(α+β)) wherex, y∈R²;e^iα,e^iβ ∈S¹.

This group may be identified with a matrix subgroup ofGL(2,C)via the map (x,e^iα)→

e^iα x 0 1

.

Unitary irreducible representations of M(2)are completely described by Mackey’s theory of induced representations. For anyξ ∈R²andg∈M(2),we defineU_g^ξas follows:

U_g^ξF (s)=e^ix,sξF (r⁻¹s) forg=(x, r)andF ∈L²(S¹).

It is known thatU_ξ is equivalent toU_ξ iff|ξ| = |ξ|.The above collection gives all the unitary irreducible representations ofM(2)sufficient for the Plancherel theorem to be true. The Plancherel theorem (see Theorem 4.2 of [10]) reads

M(2)|f (g)|²dg=

R² ˆf (ξ)²HSdξ

where f (ξ)ˆ is the ‘group Fourier transform’ defined as an operator from L²(S¹) to L²(S¹)by

f (ξ)ˆ =

M(2)f (g)U_g^ξdg.

Moreover, the group Fourier transformf (ξ),ˆ forξ ∈R²off ∈L¹(M(2))is an integral operator with the kernelk_f(ξ,e^iα,e^iβ)where

kf(ξ,e^iα,e^iβ)= ˜f (e^iβξ,e^i(β−α)),

andf˜is the Euclidean Fourier transform off in theR²-variable.

The Lie algebra ofM(2)is given by ^{iα x}_{0 0}

:(x,e^iα)∈M(2) .Let X₁=

i 0 0 0

, X₂= 0 1

0 0

, X₃= 0 i

0 0

.

Then it is easy to see that{X1, X2, X3}forms a basis for the Lie algebra ofM(2).The

‘Laplacian’_M(2)=is defined by = −(X²₁+X₂²+X²₃).

A simple computation shows that= −_R2 −_∂α^∂22 where_R2 is the Laplacian onR² given by_R2 = _∂x^∂22 +_∂y^∂22.Since_R2 and _∂α^∂2₂ commute, it follows that the heat kernel ψ_t associated to_M(2) is given by the product of the heat kernelsp_t onR² andq_t on SO(2).In other words,

ψt(x,e^iα)=pt(x)qt(e^iα)= 1 4πte^−|x|

2

4t

n∈Z

e^−n2te^inα.

Letf ∈L²(M(2)).Expandingf in theSO(2)variable we obtain f (x,e^iα)=

m∈Z

fm(x)e^imα,

wherefm(x)= _2π¹ π

−πf (x,e^iα)e^{−im α}dαand the convergence is understood in theL²- sense. Sincep_t is radial (as a function onR²) a simple computation shows that

f ∗ψ_t(x,e^iα)=

m∈Z

f_m∗p_t(x)e^−m2te^imα.

LetC^∗ =C\{0}andH(C²×C^∗)be the Hilbert space of holomorphic functions on C²×C^∗which are square integrable with respect toμν(z, w)where

dμ(z)= 1 2πte^−|y|

2

2t dxdyonC² and

dν(w)= 1 2π

√1 2πt

e^−(ln|w|)

2 2t

|w|² dwonC^∗.

Using the Segal–Bargmann result forR²andS¹we can easily prove the following theorem:

Theorem 1.1. If f ∈ L²(M(2)), then f ∗ψ_t extends holomorphically to C²×C^∗. Moreover, the mapf →f ∗ψt is a unitary map fromL²(M(2))ontoH(C²×C^∗).

2. Generalizations of Segal–Bargmann transform

In [3], Hall had proved the following generalizations of the Segal–Bargmann transform forRand compact Lie group:

Theorem 2.1.

(I) Letμbe any measurable function onRⁿsuch that

• μis strictly positive and locally bounded away from zero,

• ∀x ∈Rⁿ, σ (x)=

Rⁿe^2x·yμ(y)dy <∞. Define, forz∈Cⁿ,

ψ(z)=

Rⁿ

e^ia(y)

√σ (y)e^−iy·zdy,

whereais a real-valued measurable function onRⁿ.Then the mappingC_ψ:L²(Rⁿ)→ O(Cⁿ)defined by

C_ψ(z)=

Rⁿf (x)ψ(z−x)dx

is an isometric isomorphism ofL²(Rⁿ)ontoO(Cⁿ)L²(Cⁿ,dxμ(y)dy).

(II) LetKbe a compact Lie group andGbe its complexification. Letνbe a measure on Gsuch that

• νis bi-K-invariant,

• νis given by a positive density which is locally bounded away from zero,

• For each irreducible representationπofK,analytically continued toG, δ(π)= 1

dimV_π

Gπ(g⁻¹)²dν(g) <∞. Defineχ(g)=

π∈ ˆK dim√δ(π)VπTr(π(g⁻¹)U_π)whereg∈GandU_π’s are arbitrary unitary matrices.

Then the mapping Cχf (g)=

Kf (k)χ(k⁻¹g)dk is an isometric isomorphism ofL²(K)onto

O(G)

L²(G,dν(w)).

In this section, we shall prove an analogue of the above theorem forM(2).

Letμbe any radial real-valued function onR²such that it satisfies the conditions of Theorem 2.1(I). Define, forz∈C²

ψ(z)=

R²

e^ia(y)

√σ (y)e^−iy.zdy,

whereais a real-valued measurable function onR².Next, letνbe a measure onC^∗such that

• νisS¹-invariant,

• νis given by a positive density which is locally bounded away from zero,

• ∀n∈Z, δ(n)=

C^∗|w|²ⁿdν(w) <∞. Defineχ(w)=

n∈Z√cδ(n)n wⁿforw ∈C^∗andcn ∈ Csuch that|cn| =1.Also define φ(z, w)=ψ(z)χ(w)forz∈C², w∈C^∗.It is easy to see thatφ(z, w)is a holomorphic function onC²×C^∗.We have the following Paley–Wiener type theorem.

Theorem 2.2. The mapping C_φf (z, w)=

M(2)f (ξ,e^iα)φ((ξ,e^iα)⁻¹(z, w))dξdα is an isometric isomorphism ofL²(M(2))onto

O(C²×C^∗)

L²(C²×C^∗, μ(y)dxdydν(w)).

Proof. Letf ∈L²(M(2))and f (x,e^iα)=

m∈Z

fm(x)e^imα (2.1)

wheref_m(x)= _2π¹ _π

−πf (x,e^iα)e^−imαdα.Since the functionφ(x,e^iα),for(x,e^iα) ∈ M(2)is radial in theR²variablex,a simple computation shows that the Fourier series of f ∗φ(x,e^iα)is given by

f ∗φ(x,e^iα)=

m∈Z

fm∗ψ(x) c_m

√δ(m)e^imα.

Now, notice thatf_m∈L²(R²), ∀m∈Zandf_m∗ψis a holomorphic function onC². Moreover, by Theorem 8 of [3] we have

C²|f_m∗ψ(z)|²μ(y)dxdy=

R²|f_m(x)|²dx. (2.2)

Naturally, the analytic continuation off∗φ(x,e^iα)toC²×C^∗is given by f ∗φ(z, w)=

m∈Z

f_m∗ψ(z) cm

√δ(m)w^m. (2.3)

We show that the series in (2.3) converges uniformly on compact sets inC²×C^∗proving the holomorphicity. LetKbe a compact set inC²×C^∗.For(z, w)∈K, we have

m∈Z

f_m∗ψ(z) cm

√δ(m)w^m ≤

m∈Z

|f_m∗ψ(z)|√|w|^m

δ(m). (2.4)

By Fourier inversion (see also Theorem 8 in [3]) f_m∗ψ(z)=

R²

f_m(ξ) e^ia(ξ)

√σ (ξ)e^−iξ(x+iy)dξ,

wherez =x +iy ∈ C²andf_mis the Fourier transform off_m.Hence, ifzvaries in a compact subset ofC²,we have

|f_m∗ψ(z)| ≤ f_m2

R²

e^2ξ·y σ (ξ)dξ

¹₂

≤Cfm2.

Using the above in(2.4)and assuming|w| ≤R(aswvaries in a compact set inC^∗) we have

m∈Z

f_m∗ψ(z) c_m

√δ(m)w^m

≤C

m∈Z

f_m2

R^m

√δ(m).

Applying Cauchy–Schwarz inequality to the above, noting that

m∈Z

fm²2= f²2 and

m∈Z

R^2m δ(m) <∞

we prove the above claim. Applying Theorem 10 in [3] forS¹, we obtain

C^∗|f ∗φ(z, w)|²dν(w)=

m∈Z

|f_m∗ψ(z)|².

Integrating the above againstμ(y)dxdyonC²and using (2.2) we obtain that

C²

C^∗|f ∗φ(z, w)|²μ(y)dxdydν(w)= f²₂.

To prove that the mapC_φis surjective it suffices to prove that the range ofC_φis dense inO(C²×C^∗)∩L²(C²×C^∗, μ(y)dxdydν(w)).For this, consider functions of the form f (x,e^iα)=g(x)e^imα∈L²(M(2))whereg∈L²(R²).Then a simple computation shows that

f ∗φ(z, w)=g∗ψ(z)w^mfor(z, w)∈C²×C^∗.

SupposeF ∈O(C²×C^∗)∩L²(C²×C^∗, μ(y)dxdydν(w))be such that

C²×C^∗F (z, w)g∗ψ(z)w¯^mμ(y)dxdydν(w)=0, (2.5) for allg∈L²(R²)and for allm∈Z.From (2.5) we have

C^∗

C²F (z, w)g∗ψ(z)dμ(z)

¯

w^mdν(w)=0, which by Theorem 10 of [3] implies that

C²F (z, w)g∗ψ(z)dμ(z)=0.

Finally, an application of Theorem 8 of [3] shows thatF ≡0.Hence the proof. 2

3. Poisson integrals and Paley–Wiener type theorems

In this section we study the Poisson integrals on M(2). We also find conditions on the ‘group Fourier transform’ of a function so that it extends holomorphically to an appro- priate domain in the complexification of the group. We start with the following Gutzmer- type lemma:

Lemma 3.1. Letf ∈L²(M(2))extend holomorphically to the domain

(z, w)∈C²×C^∗:|Imz|< t, 1

R <|w|< R

and

sup

|y|<s,_N¹<|w|<N

M(2)|f (x+iy,|w|e^iθ)|²dxdθ <∞ for alls < tandN < R.Then

M(2)|f (x+iy,|w|e^iθ)|²dxdθ=

n∈Z

R²|f_n(ξ)|²e^−2ξ·ydξ

|w|²ⁿ

provided|y|< tand _R¹ <|w|< R.Conversely, if sup

|y|<s,_N¹<|w|<N

n∈Z

R²|f_n(ξ)|²e^−2ξ·ydξ

|w|²ⁿ<∞ ∀s < tandN < R

thenf extends holomorphically to the domain

(z, w)∈C²×C^∗:|Imz|< t, 1

R <|w|< R

and

sup

|y|<s,_N¹<|w|<N

M(2)|f (x+iy,|w|e^iθ)|²dxdθ <∞ ∀s < tandN < R.

Proof. Notice thatfn(x)= _2π¹ _π

−πf (x,e^iα)e^−inαdα.It follows thatfn(x)has a holo- morphic extension to{z∈C²:|Imz|< t}and

|supy|<s

R²|f_n(x+iy)|²dx <∞ ∀s < t.

Consequently,

R²|fn(x+iy)|²dx =

R²|fn(ξ)|²e^−2ξ·ydξfor|y|< s∀s < t.

Now, for each fixedz∈C²with|Imz|< sthe functionw→f (z, w)is holomorphic in the annulus{w∈C^∗:_N¹ <|w|< N}for everys < tandN < Rand so admits a Laurent series expansion

f (z, w)=

m∈Z

am(z)w^m.

It follows thatam(z) = fm(z)∀m ∈ Z.The first part of the lemma is proved now by appealing to the Plancherel theorem onS¹andR².Converse can also be proved similarly.

Recall from the Introduction that the Laplacian onM(2)is given by= −_R2

−_∂α^∂22.Iff ∈L²(M(2))it is easy to see that e^−t

1

2f (x,e^iα)=

m∈Z

R²

f_m(ξ)e^{−t (|ξ|2+m2)12}e^ix·ξdξ

e^imα.

We have the following (almost) characterization of the Poisson integrals. Let_s denote the domain inC²×C^∗defined by

_s = {(z, w):|Imz|< s,e^−s <|w|<e^s}. Theorem 3.2. Letf ∈L²(M(2)).Theng=e^−t

1

2f extends to a holomorphic function on the domain_√^t

2 and

sup

|y|<√^t

2,e^−√t2<|w|<e^√t2

M(2)|g(x+iy,|w|e^iα)|²dxdα <∞. Conversely, letgbe a holomorphic function on_t and

{|y|<s,esup^−s<|w|<e^s}

M(2)|g(x+iy,|w|e^iα)|²dxdα <∞fors < t.

Then∀s < t, ∃f ∈L²(M(2))such that e^−s

1 2f =g.

Proof. We know that, iff ∈L²(M(2))then g(x,e^iα)=e^−t

1

2f (x,e^iα)=

m∈Z

R²

fm(ξ)e^{−t (|ξ|2+m2)12}e^ix·ξdξ

e^imα.

Also,g(x,e^iα)=

m∈Zg_m(x)e^imα withg_m(ξ)=f_m(ξ)e^{−t (|ξ|2+m2)12}. Ifs ≤ ^√t

2 it is easy to show that

{^ξ∈Rsup²,m∈Z}e^{−2t (|ξ|2+m2)}

1

2e^2|ξ||y|e^2|m|s ≤C <∞for|y| ≤ t

√2.

It follows that

sup

|y|<√^t

2,e^−√t2<|w|<e^√t2

m∈Z

R²|g_m(ξ)|²e^−2ξ·ydξ

|w|^2m<∞.

By the previous lemma we prove the first part of the theorem.

Conversely, letgbe a holomorphic function on_tand sup

{|y|<s,e^−s<|w|<e^s}

M(2)|g(x+iy,|w|e^iα)|²dxdα <∞fors < t.

By Lemma 3.1 we have sup

{|y|<s,e^−s<|w|<e^s}

n∈Z

R²|g_n(ξ)|²e^−2ξ·ydξ

|w|²ⁿ<∞fors < t.

Integrating the above over|y| =s < t,we obtain

n∈Z

R²|g_n(ξ)|²J₀(i2s|ξ|)dξ

|w|²ⁿ<∞,

whereJ₀is the Bessel function of first kind. Noting thatJ₀(i2s|ξ|)∼e^2s|ξ|for large|ξ| we obtain

n∈Z

R²|g_n(ξ)|²e^2s|ξ|e^2|n|sdξ <∞fors < t.

This surely implies that

n∈Z

R²|g_n(ξ)|²e^2s(|ξ|2+m2)

1

2dξ <∞fors < t.

Definingf_m(ξ)byf_m(ξ)=g_m(ξ)e^{s(|ξ|2+m2)21} we obtain f (x,e^iα)=

m∈Z

f_m(x)e^imα∈L²(M(2))

andg=e^−s

1 2f.

Remark 3.3. A similar result may be proved for the operator e^−t

1 2 R2e^−t

1 2 S1.

It is known that a functionf ∈L²(R)extends holomorphically to the complex plane Cwith

R|f (x+iy)|²dx <∞, ∀y∈R if and only if

Re^s|ξ|| ˜f (ξ)|²dξ <∞, ∀s∈R

wheref˜denotes the Fourier transform off.The second condition is the same as

R|e^i(x+iy)ξ|²| ˜f (ξ)|²dξ <∞, ∀y ∈R.

Hereξ →e^i(x+iy)ξ may be seen as the complexification of the parameters of the unitary irreducible representationsξ → e^ixξ ofR.This point of view was further developed by Roe Goodman (see Theorem 3.1 of [2]). We shall prove an analogue of the above theorem in the case ofM(2).

Analytic vectors

Letπbe an unitary representation of a Lie groupGon a Hilbert spaceH.A vectorv∈H is called an analytic vector forπif the functiong→π(g)vis analytic.

Recall the representationsUg^ξfrom the Introduction. Denote byU_g^athe representations Ug^(a,0)fora >0.Ifen(θ)=e^inθ ∈L²(S¹),it is easy to see thaten’s are analytic vectors for these representations. Forg=(x,e^iα)∈M(2),we have

(U_g^ae_n)(θ)=e^ix,aeiθe^in(θ−α).

This action ofU_g^aone_ncan clearly be analytically continued toC²×C^∗and we obtain (U_(z,w)^a e_n)(θ)=e^ix,aeiθe^−y,aeiθw⁻ⁿe^inθ

where(z, w)∈C²×C^∗andz=x+iy∈C².

We also note that the action ofS¹onR²naturally extends to an action ofC^∗onC² given by

w(z1, z2)=(z1cosζ −z2sinζ, z1sinζ +z2cosζ ),

wherew=e^iζ ∈C^∗and(z₁, z₂)∈C².Then we have the following theorem:

Theorem 3.4. Letf ∈L²(M(2)).Thenf extends holomorphically toC²×C^∗with

|y|=r

M(2)|f (w⁻¹(x+iy),|w|e^iα)|²dxdαdσ_r(y) <∞

(whereσ_ris the normalized surface area measure on the sphere{|y| =r} ⊆R²) iff _∞

0

|y|=rU_(z,w)^a f (a)ˆ ²_HSdσ_r(y)ada <∞ wherez=x+iy∈C²andw∈C^∗.In this case we also have

_∞

0

|y|=rU_(z,w)^a f (a)ˆ ²_HSdσ_r(y)ada

=

|y|=r

M(2)|f (w⁻¹(x+iy),|w|e^iα)|²dxdαdσ_r(y).

Proof. First assume thatf ∈L²(M(2))satisfies the transformation property

f (e^iθx,e^iα)=e^imθf (x,e^iα) (3.1)

for some fixedm∈Zand∀(x,e^iα)∈M(2).As earlier we have (f (a)eˆ _n)(θ)=f_n(ae^iθ)e^inθ.

By the Hecke–Bochner identity, we have

f_n(ae^iθ)=i^−|m|a^|m|(F2+2|m|g)(a)e^imθ

whereF2+2|m|(g)is the 2+2|m|-dimensional Fourier transform ofg(x)=^f_|ⁿ_x⁽_|^|_|^xm^||⁾,consi- dered as a radial function onR^2+2|m|.

Hence,

(f (a)eˆ n)(θ)=i^−|m|a^|m|(F2+2|m|g)(a)e^i(m+n)θ.

It follows thatf (a)eˆ _nis an analytic vector and we can applyU_(z,w)^a to the above. We obtain (U_(z,w)^a f (a)eˆ _n)(θ)

=e^ix,aeiθe^−y,aeiθi^−|m|a^|m|(F2+2|m|g)(a)w^−(m+n)e^i(m+n)θ. Thus,

S¹|[U_(z,w)^a f (a)eˆ n](θ)|²dθ

= |w|^−2(m+n)

S¹a^2m|(F2+2mg)(a)|²e^−2y,aeiθdθ

= |w|^−2(m+n)

S¹e^−2y,aeiθ|fn(ae^iθ)|²dθ.

Hence,

_∞

0

U_(z,w)^a f (a)ˆ ²_HSada

= |w|^−2m

n∈Z

|w|⁻²ⁿ

R²e^−2y,ξ|fn(ξ)|²dξ

. (3.2)

Notice that, if f extends holomorphically toC²×C^∗ we must have f (w⁻¹z, w) = w^−mf (z, w)∀(z, w)∈C²×C^∗because of (3.1).

In view of Lemma 3.1, the above remark and the identity (3.2), the theorem is established for functions with transformation property (3.1) and we obtain

_∞

0

|y|=rU_(z,w)^a f (a)ˆ ²_HSdσ_r(y)ada

=

|y|=r

M(2)|f (w⁻¹(x+iy),|w|e^iα)|²dxdαdσr(y). (3.3) Next, we deal with the general case. Forf ∈L²(M(2))define

f^m(x,e^iα)=

S¹f (e^iθx,e^iα)e^−imθdθ.

Thenf^m(e^iθx,e^iα)=e^imθf^m(x,e^iα)andf_m’s are orthogonal onM(2).Assume thatf extends holomorphically toC²×C^∗.Then, so doesf^mfor allm∈Zand we have

|y|=r

M(2)|f (w⁻¹(x+iy),|w|e^iα)|²dxdαdσr(y)

=

m∈Z

|y|=r

M(2)|f^m(w⁻¹(x+iy),|w|e^iα)|²dxdαdσ_r(y). (3.4) This follows from the fact that

|y|=r

R²f^m(w⁻¹(x+iy), w)f^l(w⁻¹(x+iy), w)dxdσ_r(y)=0 ifm=l.

Applying identity (3.3) we get from (3.4)

m∈Z

_∞

0

|y|=rU_(z,w)^a f^m(a)²HSadadσ_r(y) <∞. (3.5) Now, letT , SHS=

n∈ZT e_n, Se_ndenote the inner product in the space of Hilbert–

Schmidt operators onL²(S¹).Then we notice that

|y|=rU_(z,w)^a f^m(a), U_(z,w)^a f^l(a)HSdσ_r(y)

=δ_ml

n∈Z

(−1)^−la i

m+l

(F2+2|m|g^m)(a)

×(F2+2|l|g^l)(a)w^−(m+n)(w)^−(l+n)J0(2ira). (3.6) Hence (3.5) implies that

_∞

0

|y|=rU_(z,w)^a f (a)ˆ ²_HSadadσ_r(y) <∞.

To prove the converse, we first show thatf has a holomorphic extension to whole of C²×C^∗.Recall that we have

(f (a)eˆ n)(θ)=fn(ae^iθ)e^inθ. Expandingf_n(ae^iθ)into Fourier series we have

fn(ae^iθ)=

k∈Z

Ca,n(k)e^ikθ.

Hence(U_(z,w)^a f (a)eˆ _n)(θ)is given by

k∈Z

C_a,n(k)e^ix,aeiθe^−y,aeiθw^−(k+n)e^i(k+n)θ.

Thus

|y|=r

S¹|[U_(z,w)^a f (a)eˆ _n](θ)|²dθdσ_r(y)=J₀(2ira)

k∈Z

|C_a,n(k)|²|w|^−(k+n).

Notice that J0(2ira) ∼ e^2ra for large a and

k∈Z|Ca,n(k)|²=

S¹|fn(ae^iθ)|²dθ.

If e^−r <|w|<e^r,we obtain

n∈Z

_∞

0

S¹|f_n(ae^iθ)|²e^2radθada

|w|²ⁿ<∞,

which implies

n∈Z

R²|fn(ξ)|²e^2r|ξ|dξ

|w|²ⁿ<∞.

Since this is true for allr >0 andw∈C^∗,by Lemma 3.1f extends holomorphically to C²×C^∗.It follows thatf^m(x,e^iα)defined by

f^m(x,e^iα)=

S¹f (e^iθx,e^iα)e^−imθdθ also extends holomorphically toC²×C^∗and

M(2)|f^m((x+iy),|w|e^iα)|²dxdα <∞.

Now the proof can be completed using the identity (3.3), orthogonality ofU_(z,w)^a f^m(a) (see (3.6)) and (3.4).

4. A Paley–Wiener theorem for the inverse Fourier transform

Recall from the Introduction that the ‘group Fourier transform’f (ξ),ˆ forξ ∈R²off ∈ L¹(M(2))is an integral operator with the kernelk_f(ξ,e^iα,e^iβ)wherek_f(ξ,e^iα,e^iβ)= f (e˜ ^iβξ,e^i(β−α)),f˜being the Euclidean Fourier transform offin theR²-variable. We have the following Paley–Wiener theorem for the inverse Fourier transform:

Theorem 4.1. Letf ∈L¹(M(2))be such thatf (ξ)ˆ ≡0 for all|ξ| > Rand the kernel k_f off (ξ)ˆ is smooth onR²×S¹×S¹.Thenx →f (x,e^iα)extends to an entire function of exponential typeRsuch that

sup

e^iα∈S¹|z^mf (z,e^iα)| ≤c_me^R|Imz|, ∀z∈C², ∀m∈N². (4.1) Conversely, iff extends to an entire function onC²in the first variable and satisfies(4.1) thenf (ξ)ˆ =0 for|ξ|> Randk_f is smooth onR².

Proof. We have

(f (ξ)F )(eˆ ^iα)=

S¹kf(ξ,e^iα,e^iβ)F (e^iβ)dβ, forF ∈L²(S¹) wherek_f(ξ,e^iα,e^iβ)= ˜f (e^iβξ,e^i(β−α)).

Assume thatf (ξ)ˆ ≡0 for all|ξ|> R.Sincek_f is smooth we havef (˜·,e^iα)∈C^∞_c (R²).

By the Paley–Wiener theorem onR²we obtain thatf (·,e^iα)extends to an entire function onC²of exponential type. Moreover, ifm∈N²,

z^mf (z,e^iα)=

|ξ|≤R

∂^mf˜

∂ξ^m(ξ,e^iα)e^iξ·zdξ.

It follows that sup

e^iα∈S¹|z^mf (z,e^iα)| ≤ sup

e^iα∈S¹

∂^mf˜

∂ξ^m(·,e^iα)

1

e^R|Imz|.

Conversely, iff is holomorphic onC²and satisfies (4.1), by the Paley–Wiener theorem onR²we getf (˜·,e^iα)∈C_c^∞(R²)andf (ξ,˜ e^iα)=0 for all|ξ|> R.Hencef (ξ)ˆ ≡0 for all|ξ|> R.Moreover,kf is smooth onR²sincef˜is smooth.

Acknowledgements

We thank the referee for his careful reading and useful suggestions which helped in improv- ing the presentation of the paper. The first author was supported in part by a grant from UGC via DSA-SAP and the second author was supported by Shyama Prasad Mukherjee Fellowship from Council of Scientific and Industrial Research, India.

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