An analogue of the Wiener Tauberian Theorem for the Heisenberg Motion group

1Stat-Math Unit, Indian Statistical Institute, 203 B.

T. Rd., Calcutta 700108, India

2Department of Mathematics, Indian Institute of Science, Bangalore 560012, India

rudra@isical.ac.in veluma@math.iisc.ernet.in

Mathematics Subject Classification 2000: 43A80, 22E

Keywords:Wiener Tauberian Theorem, Heisenberg Motion group.

REVIEWS

An analogue of the Wiener Tauberian Theorem for the Heisenberg Motion group

Rudra P Sarkar

AND Sundaram Thangavelu

Abstract|We show that the Wiener Tauberian property holds for the Heisenberg Motion groupTⁿB<Hⁿ. This is a special case of the same result for a wider class of groups. However, our exposition is almost self contained and the techniques used in the proof are relatively simple.

1. Introduction

A locally compact group is said to have Wiener’s property if every two sided ideal in the Banach

∗algebraL¹(G)is contained in the kernel of a nondegenerate Banach∗representation ofL¹(G) on a Hilbert space. It is well known from the work of Leptin [7] that semidirect product of abelian groups and connected nilpotent groups have the Wiener’s property. On the other hand it was established by M. Duflo that no semisimple Lie group has this property [7].

IfGis compact extension of a nilpotent group then it it is of polynomial growth [4] andGhas a symmetric group algebra. These two properties together imply thatGhas Wiener’s property [8]. For the case of general locally compact motion groups see the work of Gangolli [3].

We consider one such groupG=TⁿB<Hⁿ, the semidirect product of then-dimensional torus Tⁿand 2n+1 dimensioanl Heisenberg groupHⁿ. The above theory points out that it has Wiener’s property. However this far reaching general theory related to the groups of polynomial growth involves heavy machinary and hence not easily accessible.

We offer here a direct and independent proof of the fact forGas above, starting from a result of Hulanicki and Ricci [5]. We interpret the result in [5] onHⁿas a Wiener’s theorem forTⁿ-biinvariant functions on G. Using elementary arguments

we extend this result to a Wiener’s theorem for the full group G. Going towards the proof we find the representations in bG and establish the Plancherel theorem. Then we obtain the Wiener’s theorem explicitly in terms of the representations.

Precisely, we find the sufficient condition that an L¹-function onGgenerates a dense ideal inL¹(G).

This condition is also necessary. We may conjecture that this method of extending the theorem from the biinvariant functions inL¹(G)to the fullL¹(G) would work for other compact extensions.

This exposition is almost self contained and techniques are simple. Our groupGis a subgroup of the so called Heisenberg Motion group, which is the semidirect product of the Heisenberg group Hⁿand the unitary groupU(n). This latter group acts onHⁿas automorphisms and the semidirect productU(n)B<Hⁿturns out to be the natural group of isometries for the Heisenberg geometry;

see the works of Koranyi [6] and Strichartz [10]. In [9] R. Rawat has studied a Wiener’s theorem for the action ofU(n)B<HⁿonHⁿ.

2. Notation and Preliminaries LetHⁿ=Cⁿ×R, with group law

(z,t)◦(z⁰,t⁰)=

z+z⁰,t+t⁰−1 2I m z⁰.z

,

denote the (2n+1)-dimensional Heisenberg group. LetTⁿ=S¹× ··· ×S¹(ntimes) be then- dimensioanl Torus. The groupTⁿacts naturally onHⁿby automorphisms and(Hⁿ,Tⁿ)forms a Gelfand pair. LetG=TⁿB<Hⁿbe the semidirect product ofHⁿandTⁿ. Let us denote the elements ofGby(σ,z,t)whereσ=(e^iθ1,...,e^iθn)∈Tⁿand (z,t)∈Hⁿ. The group law ofGis given by:

(σ,z,t).(τ,w,s)

=

στ,z+σw,t+s−1

2I mσw.¯z

. AsTⁿis a subgroup ofGthrough the identification ofσand(σ,0,0), it acts onGfrom left and right through the group law as:

(σ,0,0).(τ,w,s)=(στ,σw,s) and

(τ,w,s).(σ,0,0)=(στ,w,s).

The Heisenberg group Hⁿ can also be identified naturally both as a subgroup of G and as the quotient G/Tⁿ. For an element (σ,z,t)∈G, (σ,z,t)⁻¹=(σ⁻¹,−σ⁻¹z,−t)= (σ⁻¹,0,0)(1,−z,−t).

We use Rⁿ+ for R+× ··· ×R+ (n times) where R+ is the set of positive reals. The n- dimesional Euclidean space is denoted by Rⁿ. For this article ‘.’ is the usual bilinear inner product. For2=(θ1,...θn)∈ [0,2π)ⁿwe denote (e^iθ1, ... ,e^iθn)∈Tⁿ by eⁱ². A function f on Hⁿis calledpolyradialiff(z,t)=f(eⁱ²z,t)for all eⁱ²∈Tⁿ, where z=(z1, ... ,z_n)∈Cⁿ and eⁱ²z=(e^iθ1z₁,...,e^iθnz_n). We considerC^∞_c (G)to be the space of compactly supportedC^∞-functions.

For a functionf∈C_c^∞(G)we define a function f₀ onHⁿbyf₀(z,t)=f(1,z,t)and a polyradial functionf00onHⁿbyf00(r,t)=f(1,r,t), where r=(r₁,...,r_n),r_j≥0. Forµ=(µ1,...,µn)∈Zⁿ the characterχ_µ ofTⁿ is defined byχ_µ(eⁱ²)= e^µ.2=e^iµ1θ1.···.e^iµnθn. Suppose µ,µ⁰∈Zⁿ. A complex valued functionf onGis called spherical of right (resp. left) typeµ(resp.µ⁰) if

f(xk1) = f(x)χ_µ(k1) (resp.f(k2x)

= χ_µ⁰(k2)f(x))

for allx∈G and k₁,k₂∈Tⁿ. A function of left typeµ⁰and right typeµis called spherical of type (µ⁰,µ). For a functionf∈C^∞_c (G)

Z

Tⁿχµ(k₁)f(xk₁)dk₁ and Z

Tⁿχ_µ⁰(k₂)f(k₂x)dk₂

respectively are its projections on the space of right µ and left µ⁰ type functions in C^∞_c (G). For a suitable function space S , by S_µ⁰_,µ we denote the projection of S on the subspace of left µ⁰ and right µtype functions ofS. The polyradial functions inL¹(Hⁿ)(denoted byAin [5]) is under obvious identification the same asL¹(G)_0,0, the bi- Tⁿinvariant functions inL¹(G). The right and left G-translates off forx∈G, are denoted respectively by f^x and ^xf. Precisely, f^x(y)=f(yx⁻¹) and

xf(y)=f(x⁻¹y). By∗we mean convolution in Gwhile∗_Hndenotes the convolution inHⁿ. For two elementsm,n∈Rⁿ,m−nis the component- wise subtraction. Bym>nwe meanm_j>n_jfor everyj=1,...,n.

We conclude this section with the following proposition.

Proposition 2.1. Let f,g∈C^∞_c (G). Then, (i) left type off∗gis the left type off and the right

type of f∗g is right type of g.

(ii) if moreoverf andgare of rightm-type and left n-type respectively, then,f∗g≡0ifm6=nand if m=n then f∗g(1,z,t)=f₀∗_Hng₀(z,t) Proof of this proposition follows easily considering the fact that every elementg∈Gcan be decomposed asg=xk=k₁x₁wherex,x₁∈Hⁿandk,k₁∈Tⁿ. By (i) aboveL¹(G)_µ,µ is a subalgebra ofL¹(G) under convolution∗.

3. Representations ofG

We shall construct the representations ofGfrom that ofHⁿand the Euclidean motion groupM(2). For details of the representations of these two groups we refer to [12] and [11] respectively.

For k=0,1,2, ... , and t∈R the Hermite polynomials are defined by

H_k(t)=(−1)^k d^k

dt^k{e^−t2}e^t2

! .

The normalized Hermite functions are defined in terms of the Hermite polynomials as

h_k(t)=(2^k√

πk!)⁻¹²H_k(t)e^−12t2. These Hermite functions{h_k:k=0,1,2,...}form an orthonormal basis ofL²(R). For any multiindex αand x∈Rⁿwe define the higher dimensional Hermite functions8_αby taking tensor product:

8_α(x)=5ⁿ_j=0h_αj(xj).

Then the family{8_α}is an orthonormal basis of L²(Rⁿ). For^l6=0 we define thescaledHermite functions

8^l_α(x)= |l|ⁿ48_α

|l|¹2x .

We also consider

8^l_αβ(x)=(2π)⁻ⁿ²|l|ⁿ²hπ^l(z,0)8^l_α,8^l_βi, which is essentially the matrix coefficient of the Schr¨odinger representation πl at (z,0) of Hⁿ. They are the so called special Hermite functions and{8^l_αβ :α,β∈Nⁿ}is a complete orthonormal system inL²(Cⁿ). For eachσ∈Tⁿ,(z,t)7→(σz,t) is an automorphism ofHⁿ, becauseTⁿpreserves the symplectic formI m(z.w). Ifρis a representation ofHⁿ, then using this automorphism we can define another representationρ_σbyρ_σ(z,t)=ρ(σz,t) which coincides withρ at the center. Therefore by Stone-von Neumann theoremρ_σis unitarily equivalent toρ. If we takeρto be the Schr¨odinger representation πl, then we have the unitary intertwining operatorµl(σ), i.e.

πl(σz,t)=µl(σ)πl(z,t)µl(σ)^∗. The operator valued functionµ^l can be chosen so that it becomes a unitary representation of the double cover of the symplectic group and is called metaplectic representation. For a detailed description of these representations we refer to [2].

For each ^l6=0,m∈ Zⁿ we consider the representations(ρ^l_m,L²(Rⁿ))defined by

ρ^l_m(eⁱ⁸,z,t)=e^−im.8πl(z,t)µl(eⁱ⁸), whereµ^l(eⁱ⁸)is the metaplectic representation andπ^l is the Schr¨odinger representation ofHⁿ. The action of µl(eⁱ⁸) on the Hermite basis {8^l_α:α∈Nⁿ}is given by

µl(eⁱ⁸)8^l_α=e^iα.88^l_α.

Sinceρ^l_m(1,z,t)=πl(z,t), these representations are irreducible.

Theorem 3.1. Letπbe any unitary representation of Gsuch thatπ(1,z,t)is irreducible as a representation of Hⁿ. If π(1,0,t)=e^iltI with ^l6=0 then π is unitarily equivalent toρ^l_mfor some m∈Zⁿ. Proof.Sinceπ(1,z,t)is irreducible andπ(1,0,t)= e^iltIby Stone-von Neumann theoremπis unitarily equivalent toπl(z,t)onL²(Rⁿ). IfHis the Hilbert

space on which πis realised, we have a unitary operatorUl:H−→L²(Rⁿ)such that

U_l^∗π^l(z,t)Ul=π(1,z,t).

Now

π(eⁱ⁸,z,t)=π(1,z,t)π(eⁱ⁸,0,0) and also

π(eⁱ⁸,z,t) = π((eⁱ⁸,0,0)(1,e⁻ⁱ⁸z,t))

= π(eⁱ⁸,0,0)π(1,e⁻ⁱ⁸z,t).

Therefore,

Ulπ(1,z,t)π(eⁱ⁸,0,0)Ul^∗

=Ulπ(eⁱ⁸,0,0)π(1,e⁻ⁱ⁸z,t)U_l^∗ and

πl(z,t)Ulπ(eⁱ⁸,0,0)Ul^∗

=Ulπ(eⁱ⁸,0,0)Ul^∗πl(e⁻ⁱ⁸z,t).

But πl(e⁻ⁱ⁸z,t) is unitarily equivalent to πl(z,t)via the metaplectic representation, i.e.

π^l(e⁻ⁱ⁸z,t)=µ^l(e⁻ⁱ⁸)π^l(z,t)µ(e⁻ⁱ⁸)^∗. Definingρ(eⁱ⁸)=Ulπ(eⁱ⁸,0,0)U_l^∗we have

πl(z,t)ρ(eⁱ⁸)

=ρ(eⁱ⁸)µ(e⁻ⁱ⁸)π(z,t)µl(e⁻ⁱ⁸)^∗. Thus ρ(eⁱ⁸)µ(e⁻ⁱ⁸) commutes with πl(z,t) for all (z,t) and hence ρ(eⁱ⁸)µl(e⁻ⁱ⁸)= χ(8)I. That isρ(eⁱ⁸)=χ(8)µl(eⁱ⁸). Therefore χ(8).χ(8⁰)=χ(8+8⁰). Thus χ defines a character of the group Tⁿ. Hence, ρ(eⁱ⁸)= e^−im.8µ(eⁱ⁸)for somem∈Zⁿ. Finally

Ulπ(eⁱ⁸,z,t)Ul^∗=πl(z,t)e^−im.8µ(eⁱ⁸) which proves the theorem.

We now consider the case when^l=0. That is π(eⁱ⁸,z,t)=π(eⁱ⁸,z,0).

Definingρ(z,eⁱ⁸)=π(eⁱ⁸,z,0)we see that ρ(z,eⁱ⁸)ρ(w,eⁱ²) = π(eⁱ⁸,z,0)π(eⁱ²,w,0)

= π(eⁱ⁽²⁺⁸⁾,z+eⁱ⁸w,0) and henceρis a representation of the motion group Cⁿ×Tⁿ=M(2)× ··· ×M(2)whereM(2)=C× U(1).

It is well known (see [11]) that all the irreducible unitary representations ofM(2)are given by the following two families

(i) fora>0,ρarealised onL²(T)and defined by ρa(z,e^iφ)g(θ)=e^{iRe(aze−iθ)}g(θ−φ).

(ii) for m ∈ Z, the one-dimensional representationsχmrealised onCand defined by

χm(e^iφ)z=e^imφz.

From this we build the representations ofM(2)×

··· ×M(2)as:

(I) for a=(a1, ...a_n)∈Rⁿ+, ρa realised on L²(Tⁿ)is given by

ρa(z,eⁱ⁸)g(eⁱ²) = 5ⁿ_j=1e^iRe(ajzje

−iθj)

×g(eⁱ⁽²⁻⁸⁾), wherez=(z1,...zn)∈Cⁿ,eⁱ²∈Tⁿ. (II) form∈Zⁿ,χmrealised onCis given by

χm(eⁱ⁸)w=e^im.8w. Hence we have,

Theorem 3.2. Ifπis a unitary representation ofG such thatπ(1,0,t)=Iandπ(1,z,t)is irreducible thenπis unitarily equivalent to eitherρafor some a∈Rⁿ+orχmfor some m∈Zⁿ.

We now show that the representations ρ^l_m are enough for the Plancherel theorem. Givenf∈L¹∩ L²(G)consider the group Fourier transform

bf(^l,m) = Z

f(eⁱ⁸,z,t)ρ^l_m(eⁱ⁸,z,t)d8dzdt

= Z

f^l(eⁱ⁸,z)e^−im.8πl(z,0)µl(eⁱ⁸)

×d8dz where f^l(eⁱ⁸,z)=R

e^iltf(eⁱ⁸,z,t)dt. We can calculate the Hilbert–Schmidt operator norm of bf(^l,m)by using the Hermite basis{8^l_α}:

bf(^l,m)8^l_α = Z

f^l(eⁱ⁸,z)e^−im.8e^iα.8πl

×(z,0)8^l_αd8dz

= Z

f˜^l(m−α,z)πl(z,0)8^l_αdz where

f˜^l(z,m)= Z

f^l(z,eⁱ⁸)e^−im.8d8.

Thus

(bf(^l,m)8^l_α,8^l_β)

=(2π)ⁿ²|^l|⁻ⁿ² Z

f˜^l(z,m−α)8^l_αβ(z)dz so that

kbf(^l,m)8^l_αk²₂

=(2π)ⁿ|l|⁻ⁿX

β

| Z

f˜^l(z,m−α)8^l_αβ(z)dz|²

and

kbf(^l,m)k²_{H S} = (2π)ⁿ|l|⁻ⁿ

×X

α

X

β

Z

f˜^l×(z,m−α)8^l_αβ(z)dz

2.

This shows that X

m

kbf(^l,m)k²_{H S}=(2π)ⁿ|l|⁻ⁿ

×X

m

X

α

X

β

Z

f˜^l(z,m−α)8^l_αβ(z)dz|².

Making a change of variable in the summation over mand noting that{8^l_αβ:α,β∈Nⁿ}is an orthonormal basis forL²(Cⁿ)we obtain

X

m

kbf(^l,m)k²_{H S}

=(2π)ⁿ|^l|⁻ⁿX

m

Z

| ˜f^l(z,m)|²dz

=(2π)²ⁿ|l|⁻ⁿ Z Z

|f^l(eⁱ⁸,z)|²d8dz. Therefore,

Z X

m

kbf(^l,m)k²_{H S}

!

|^l|ⁿd^l

=(2π)²ⁿZ Z Z

|f^l(eⁱ⁸,z)|²d8dz

=(2π)²ⁿ⁺¹Z

|f(eⁱ⁸,z,t)|²d8dzdt. Theorem 3.3. (Plancherel)For f∈L¹∩L²(G)

Z

|f(eⁱ⁸,z,t)|²d8dzdt

=(2π)⁻²ⁿ⁻¹Z ∞

−∞

X

m

kbf(^l,m)k²

!

|l|ⁿd^l.

Remark 3.4. Note that if f(eⁱ⁸z,t)is right Tⁿ- invariant then(bf(^l,m)8^l_α,8^l_β)=0 unless m= α≥0. Thusbf(^l,m)=0 form_j<0 for somejand form≥0

kbf(^l,m)k²_{H S}

=(2π)ⁿ|l|⁻ⁿX

β∈Zⁿ

bf(^l,m)8^l_m,8^l_β

2.

We also use the notationρ^l_m(f)forbf(^l,m).

4. Wiener’s Theorem

We begin with a theorem of Hulanicki and Ricci [5]

Theorem 4.1. Let J be a proper closed ideal of L¹(Hⁿ/Tⁿ). Suppose for every non-zero multiplicative functional3_ψofL¹(Hⁿ/Tⁿ), given by the bounded spherical function ψ, there is a function f∈J such that

3_ψ(f)= Z

f(z,t)ψ(z,t)dzdt6=0.

Then J=L¹(Hⁿ/Tⁿ).

The actual Theorem of Hulanicki and Ricci is little stronger than this as it says thatL¹(Hⁿ)∗_HnJ is dense inL¹(Hⁿ). We have quoted the part which we are going to use. A detailed proof of this can be found in [1]. Note that in our notationL¹(Hⁿ/Tⁿ) isL¹(G)_0,0. In the language of representations this theorem can be restated as: forJas above, suppose (i) for every (^l,m)∈R\ {0} ×Zⁿ there is a

functionfl,m∈Jsuch thatρ^l_m(fl,m)6=0 (ii) for everya=(a₁,...,a_n)witha_i>0 there is

af_a∈Jsuch thatρa(f_a)6=0 (iii) there isf ∈Jsuch thatR

Hⁿf dzdt6=0.

ThenJ=L¹(G)0,0. Forz=(z₁,...,z_n)∈Cⁿwe consider itsmulti-polardecomposition

z=eⁱ²kzk =(r1e^iθ1,...,rne^iθn), whereeⁱ²∈Tⁿ,kzk =(r₁,...,r_n)andz_j=r_je^iθj forj=1,...,n. Notice that condition (ii) is then equivalent to this: the Euclidean Fourier transform of fa at a∈Rⁿ+ is not equal to zero. That is bf_a(.,t)(a)6=0.

Proposition 4.2. Letf ∈L¹(G). If right(resp. left) type off isµ(resp.µ⁰)thenbf(^l,m)≡0ifm_j<µj

(resp. m_j⁰ <µj⁰) for some j,j⁰. For f ∈L¹(G) of type (µ⁰,µ), hρ^l_m(f)8^l_α, 8^l_βi 6=0 only when µj=m_j−αjandµ⁰_j=m_j−βjfor all j.

Proof.

hρ^l_m(f)8^l_α,8^l_βi

= Z

G

f(eⁱ²,z,t)h8^l_α,ρ^l_m

×(eⁱ²,z,t)⁻¹8^l_βid2dzdt

= Z

G

f(eⁱ²,z,t)h8^l_α,ρ^l_m

×(e⁻ⁱ²,−e⁻ⁱ²z,−t)8^l_βid2dzdt

= Z

G

f(eⁱ²,eⁱ²e⁻ⁱ²z,t)h8^l_α,e^{−i(β−m)2}πl

×(−e⁻ⁱ²z,−t)8^l_βid2dzdt

= Z

G

e^iµ0.2f(1,e⁻ⁱ²z,t)e^i(β−m)2

×h8^l_α,πl(−e⁻ⁱ²z,−t)8^l_βid2dzdt. Therefore,hρ^l_m(f)8^l_α,8^l_βi =0 unlessµ⁰=m− βin which case,

hρ^l_m(f)8^l_α,8^l_βi

= Z

Hⁿ

f(1,z,t)h8^l_α,πl(−z,−t)8^l_βidzdt. The other result will follow similarly.

Lemma 4.3. Suppose forf ∈L¹(G),bf(^l,m)6=0for some^l∈R\ {0},m∈Zⁿandρa(f)6=0for some a∈Rⁿ+. Assume also thatµ∈Zⁿsatisfiesµ≤m.

Then there exist functionsgandg⁰of type(µ,µ)in the closure of span of(left and right)G-translates of f such thatbg(^l,m)6=0andρa(g⁰)6=0.

Proof.Letν=m−µ. As the operatorbf(^l,m)6=0 there existαandβsuch thathbf(^l,m)8^l_α,8^l_βi 6=0.

Sinceρ^l_mis irreducible, for everyε >0 there exist c_i∈Candx_i∈G,i=1,2,...,sfor somessuch that

s

X

i=1

c_iρ^l_m(xi)8^l_ν−8^l_α <ε.

This implies

s

X

i=1

c_iρ^l_m(f)ρ^l_m(xi)8^l_ν−ρ^l_m(f)8^l_α

<εkρ^l_m(f)k and hence

ρ^l_m(g1)8^l_ν−ρ^l_m(f)8^l_α

<εkρ^l_m(f)k whereg₁=Ps

i=1c_if^xi.

As ρ^l_m(f)8^l_α 6= 0 and kρ^l_m(f)k < ∞, ρ^l_m(g₁)8^l_ν 6=0. By the above proposition this also shows that g₁ has right µ component. Let g₂ be the right µ-th projection of g₁. Then ρ^l_m(g₁)8^l_ν =ρ^l_m(g₂)8^l_ν 6=0, because Fourier transforms of other right projections ofg₁will kill 8^l_ν. There existsγsuch thathρ^l_m(g₂)8^l_ν,8^l_γi 6=0.

Forε >0, there arec_j⁰∈Candyj∈G,j=1,2,...s⁰ for somes⁰such that

s⁰

X

j=1

c_j⁰ρ^l_m(y⁻¹_j )8^l_ν−8^l_γ <ε.

Applyingρ^l_m(g₂)^∗on both sides and using thatρ^l_m is unitary we get,

s⁰

X

j=1

c⁰_jρ^l_m(g2)^∗ρ^l_m(yj)^∗8^l_ν

−ρ^l_m(g₂)^∗8^l_γ

<εkρ^l_m(g₂)^∗k. That implies

s⁰

X

j=1

c⁰_jρ^l_m(^yjg₂)^∗8^l_ν−ρ^l_m(g₂)^∗8^l_γ

<εkρ^l_m(g₂)^∗k.

Letg₃=Ps⁰

j=1c⁰_j ^yjg₂. Then we have

kρ^l_m(g3)^∗8^l_ν−ρ^l_m(g2)^∗8^l_γk<εkρ^l_m(g2)^∗k. Asρ^l_m(g2)^∗8^l_γ6=0,ρ^l_m(g3)^∗8^l_ν6=0. Sinceg₃is a finite linear combination of the leftG-translates ofg₂, right type ofg₃ continues to beµ. Hence h8^l_ν,ρ^l_m(g3)^∗8^l_νi = hρ^l_m(g3)8^l_ν,8^l_νi 6=0. Letgbe the leftµ-th projection ofg₃. Theng is of type (µ,µ)andhρ^l_m(g3)8^l_ν,8^l_νi 6=0.

By similar method we can show that there is ag⁰of type(µ,µ)in the closure of the span of G-translates off such thatρa(g⁰)6=0.

Suppose f is a function in L¹(G)_µ,µ. Let z=eⁱ²kzkwherekzk =(r₁,...,r_n)andeⁱ²∈Tⁿ. Then

f(eⁱ⁸,z,t) = f(eⁱ⁸,eⁱ²kzk,t)

= e^iµ.8f(1,eⁱ²kzk,t).

Since(1,eⁱ²kzk,t)=(eⁱ²,0,0)(e⁻ⁱ²,kzk,t)we also have

f(1,eⁱ²kzk,t) = e^iµ.2f(e⁻ⁱ²,kzk,t)

= f(1,kzk,t).

Thus we have

f(eⁱ⁸,z,t)=e^iµ8f₀₀(kzk,t).

By the above proposition for f ∈L¹(G)_µ,µ, ρ^l_m(f)=0 wheneverm_j<µjfor somej. Ifm≥µ then

ρ^l_m(f) = Z

G

f(eⁱ⁸,z,t)ρ^l_m(eⁱ⁸,z,t)d8dzdt

= Z

G

f₀₀(kzk,t)e^iµ.8e^−im.8πl(z,t)µl

×(eⁱ⁸)d8dzdt

= Z

G

f₀₀(kzk,t)ρ^l_m−µ(eⁱ⁸,z,t)d8dzdt. This shows thatρ^l_m(f)=ρ^l_m−µ(f00).

Now suppose for somef ∈L¹(G) χm(f)=

Z

G

f(eⁱ⁸,z,t)e^−im.8dzdt d86=0. Letf_µis the rightµ-th projection off. That is

f_µ(eⁱ⁸,z,t) = Z

G

f(eⁱ⁽²⁺⁸⁾,z,t)e^−iµ.2d2

= e^iµ.8 Z

G

f(eⁱ²,z,t)e^−iµ2d2.

Then

χm(f_µ) = δm,µ

Z

G

f(eⁱ²,z,t)e^−iµ.2d2dzdt

= δ_m,µχ_µ(f)

where δm,µ is the Kronecker δ and hence if µ=mthen χ_µ(f_µ)6=0. Similarly we can show that if _µf is the left µ-th projection of f, then χm(µf)=δm,µχµ(f). Therefore ifχm(f)6=0 for allm∈Zⁿ, thenχµ(g)6=0 wheregis the(µ,µ)- the projectionf. By the above computation that meansR

Hⁿg_0,0(kzk,t)dzdt6=0.

Proposition 4.4. Let S⊂L¹(G)_µ,µSuppose (1) for every^l∈R\ {0}andm≥µ, there isfl,m∈S

such thatρ^l_m(fl,m)6=0,

(2) for all a∈R+ⁿ,ρa(fa)6=0for some function fa∈S and

(3) χµ(f)6=0for some function f∈S.

Then the span of the ideal(under convolution∗) generated by elements ofSinL¹(G)_µ,µis dense in L¹(G)_µ,µ.

Proof.LetS_0,0= {f_0,0:f∈S}. Iff∈L¹(G)µ,µ, then ρ^l_m(f)6=0 for somem≥µimpliesρ^l_m−µ(f_0,0)6=0.

Also ρa(f)6=0 implies the Euclidean Fourier transform bf_0,0(.,t)(a)6=0. Notice also that χ_µ(f) 6= 0 implies R

Hⁿf_0,0(kzk,t)dzdt 6=0.

Therefore by Theorem 4.1 span of the ideal generated by the elements of S_0,0 in L¹(G)_0,0 is dense in L¹(G)0,0. Let us take an arbitrary h∈L¹(G)µ,µ. Forε >0, there existc_j∈C,f^j∈S andg^j∈L¹(G)_0,0,j=1,2,...sfor somessuch that

X

j

cjf_0,0^j ∗_Hng^j−h_0,0

_L1(Hn)<ε. ^(4.1)

Let us define ˜

g^j(eⁱ²,z,t)=g^j(kzk,t)e^iµ.2. Clearly g˜^j∈L¹(G)µ,µ. Then,

Xc_jf^j∗ ˜g^j−h _L1(G)

= Z

G

Xc_jf^j∗ ˜g^j(eⁱ²,z,t)−h(eⁱ²,z,t)

×d2dzdt.

Sincef^j∗ ˜g^jandhare functions of type(µ,µ), the above expression equals to

Z

G

Xcj(f^j∗ ˜g^j))_0,0(kzk,t)e^iµ.2

−h_0,0(kzk,t)e^iµ.θ|d2dzdt

= Z

Hⁿ

Xc_j(f^j∗ ˜g^j))0,0(kzk,t)

−h_0,0(kzk,t)|dzdt.

Again using the fact thatf^j∗ ˜g^jandhare(µ,µ)- functions we have

(f^j∗ ˜g^j)_0,0(kzk,t) = f^j∗ ˜g^j(1,z,t)

= (f₀^j∗_Hn ˜

g^j₀)(kzk,t) by Proposition 2.1. Also,

g˜^j₀(z,t)= ˜g^j(1,z,t)=g^j(kzk,t)=g^j(z,t) and

f₀^j(z,t)=f^j(1,z,t)=f₀^j_,0(kzk,t)=f₀^j_,0(z,t).

Therefore,

(f^j∗ ˜g^j)0,0(kzk,t)=f_0,0^j ∗_Hng^j(z,t).

Hence

Xc_jf^j∗ ˜g^j−h _L1(G)

= Z

Hⁿ

Xc_j(f₀^j_,0∗_Hng˜^j)(kzk,t)

−h_0,0(kzk,t)|dzdt.

The proposition follows now from (4.1).

We now state and prove a Wiener Tauberian theorem for the action ofGon itself.

Theorem 4.5. If for a functionf ∈L¹(G),bf(^l,m) 6=0for all^l6=0,m∈Zⁿ,bf(a)6=0for alla∈Rⁿ+

andχm(f)6=0for allm∈Zⁿthen the span of the (left and right)G-translates off is dense inL¹(Gx).

Proof. Fix µ∈Zⁿ. From lemma 4.3 and the subsequent computations we see that using the given nonvanishing condition onf, we can find functions of type(µ,µ)in the closure ofG-translates off such that they satisfy the hypothesis of proposition 4.4. Hence they can generateL¹(G)_µ,µ. The proof now follows from the fact that any idealIinL¹(G) which containsL¹(G)_µ,µfor allµ∈ZⁿisL¹(G) itself.

Note that instead of a single functionf we can also take a subset S⊂L¹(G) such that Fourier transforms of the elements ofShave no common zero in the set of above representations parametrized by R^∗×Zⁿ∪R+ⁿ ∪Zⁿ, where R^∗ is the set of nonzero reals.

Received 18 September 2007; revised 03 December 2007.

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Rudra Sarkarreceived his Ph.D from Indian Statistical Institute in 1998.

After straying into computer industry for almost five years he returned to Mathematics in 2000. Currently he is an Assistant professor with the Statistics and Mathematics division of I.S.I., Kolkata.

He is a semi-simple harmonic analyst who works on Tauberian theorems and uncertainty principles.

S. Thangavelureceived his Ph.D from Princeton in 1987. After spending six years in T.I.F.R. and twelve years in I.S.I., he moved to I.I.Sc. in 2005. Recipient of B.M. Birla Science Prize (1996) and Bhatnagar award (2002), he is a Fellow of both Indian Academy of Sciences and Indian National Science Academy. He has authored three books and published more than sixty research papers.