DER-CHEN CHANG, WAYNE EBY, AND ERIC GRINBERG
Abstract. We address certain cases in which the Pompeiu transform for the Heisenberg group Hn is known to be injective, in particular the cases of a sphere and a ball, or two
balls of appropriate radii. In these cases we develop a method which provides for the reconstruction of the function f from its integral averages.
In addition, we consider these issues in connection to the Weyl calculus and group Fourier transform. We furthermore explore issues of convergence for this method of deconvolution and related issues of size of the Gelfand transform near the zero sets. Finally, given a set of deconvolvers which work for Euclidean space Cn, we investigate the issue of how to extend
the deconvolution to the Heisenberg group, giving the extension for special cases.
1. Introduction
The Pompeiu problem asks for conditions which guarantee the uniqueness of a function in terms of its integral averages. This issue may also be expressed as the issue of injectivity of a certain integral transformation. In particular, given a function f ∈ C(Rn) and a given
set S ⊂⊂ Rn, consider the integral averages
Z
σS
f (x)dµσ(x)
for all rigid motions σ of Rn, where µσ is the area measure on the set σS. The Pompeiu
problem asks whether vanishing of all of these integrals is sufficient to conclude f ≡ 0. The same question may also be asked in the context of an integral transform. Define the Pompeiu transform as
Pf (σ) = Z
σS
f (x)dx
for each σ ∈ M(n), the Euclidean motion group. The Pompeiu problem may then be considered as the issue of injectivity for the Pompeiu transfrom. When the transform is injective, it is then possible to consider the issue of inversion.
We consider the issue of inversion of the Pompeiu transform at the level of establishing a method to recover the original function f from the transformed function Pf . Note that we do not give an explicit inverse to the operator P, but we do establish a limiting procedure that allows the recovery of f from a given Pf . The problem of deconvolution is then the reconstruction of the function f from these integral averages. In the paper [7], Berenstein and Yger addressed this problem in the setting of the Euclidean space Rn. They form explicit
2000 Mathematics Subject Classification. Primary: 32A10; Secondary: 30E99.
Key words and phrases. Euclidean spaces, Heisenberg group, Laguerre functions, deconvolution, H¨ormander strong coprime condition, Pompeiu property, Gelfand transform.
The first author is partially supported by the Hong Kong RGC grants #600607 and #601410, and the competitive research grant #GX2236000 at Georgetown University.
formulas to determine the deconvolvers ν1, . . . , νn from the bounded measures µ1, . . . , µn
using the operations of derivation, integration, convolution, and summation. In this paper we address the problem of deconvolution for the Pompeiu problem in the setting of the Heisenberg group. We consider the Pompeiu problem for functions in C ∩ L∞(Hn) and
consider integration over a set S ⊂⊂ Cn× {0} ⊂ H
n, such as described in [1]. Although the
issue of deconvolution can be considered for any set S, or collection of sets, for which the Pompeiu transform is injective, we focus on two cases for which the zeros of the associated transforms are well known. The first case considers set S as a ball and a sphere of the same radius, while the second considers S as two balls of appropriate radii. The general approach established in this paper provides a method to construct sequences of deconvolvers which allows reconstruction of f through a limiting process. In addition we address the issue of when these deconvolving sequences will converge to individual deconvolvers, defined as tempered distributions. This issue of convergence relates directly to the location of the zeros of ˆµ1, . . . , ˆµn and the issues discussed below.
In Euclidean space, a large part of the problem of deconvolution is directly related to a condition known as H¨ormander’s strongly coprime condition, [16]. Consider the integral conditions as a set of convolution equations. In the case of two balls of radii r1, r2the integral
conditions may be written as
f ∗ Tr1 = 0 f ∗ Tr2 = 0
where Trj are defined by
hφ, Trji =
Z
|x|<rj
φ(x)dµj(x)
for j = 1, 2, where µj is area measure on the ball of radius rj. Now observe that the problem
of deconvolution can be solved by finding ν1, ν2 ∈ bE which satisfy the analytic Bezout equality
(1.1) Tbr1
b
ν1+ bTr2νb2 ≡ 1.
H¨ormander’s strongly coprime condition tells us that such ν1, ν2exist as compactly supported
distributions if and only if bTr1 and bTr2 satisfy the estimate, related to Paley-Weiner estimates
| bTr1(ζ)| + | bTr2(ζ)| ≥ C
1 (1 + |ζ|)Ne
−B|Imζ|
(1.2)
for some C, N, B > 0. In addition to bTr1 and bTr2 not having common zeros, this condition
describes a maximum rate of decay for a combination of bTr1 and bTr2, ensuring that where
one of these becomes zero, the other must not become too small too quickly.
In the specific case of the Pompeiu problem for two balls of distnict radii, the issue of injectivity is determined by an arithmetic condition on the radii. In this case, the transform for balls Br1 and Br2 will be injective when
r1 r2 6∈ Q(Jn 2) = { γx γy : γ ∈ R ∗ , Jn 2(x) = J n 2(y) = 0}.
When, in addition, the quotient of radii r1
r2 is poorly approximated by quotients zeros of J n 2,
then H¨ormander’s strongly coprime condition can be shown to be satisfied. It then follows that the deconvolution problem for Tr1 and Tr2 can be satisfied with compactly supported
When such ν1and ν2 can be found, this provides a solution to the problem of deconvolution
as follows. First note that Tr1∗ ν1+ Tr2∗ ν2 = δ. Then convolve the convolution expressions
f ∗ Tr1 and f ∗ Tr2 with ν1 and ν2 to obtain
f ∗ Tr1 ∗ ν1+ f ∗ Tr2 ∗ ν2 = f ∗ (Tr1∗ ν1+ Tr2 ∗ ν2)
= f ∗ δ = f.
One of the goals of this paper is to establish a similar approach to deconvolution that works in the Heisenberg setting. We develop a method that works for the two specific cases considered in Section 3, the first case of a ball and sphere of the same radius and the second case of two balls of distinct radii. In both cases a sequence of deconvolvers is presented which in the limit accomplishes the deconvolution to return the original function f . Sections 4 considers these issues in the context of the Weyl calculus. Issues of convergence for the deconvolving sequences is considered in Section 5. In the case of two balls of distinct radii, these considerations lead to an interesting interaction between H¨ormander’s strongly coprime condition, the issue of N -well approximation of the radii, and the space in which the sequences of deconvolvers will converge. Section 6 deals with the extension of a given deconvolution for the space Cn to the space Hn. We plan to pursue these issues further in
a later paper.
The inspiration for this paper comes from the desire to find deconvolvers ν1 and ν2 that
satisfy the analytic Bezout equation (1.1). When considering this issue in the setting of the Gelfand transform for the Heisenberg group, we realized that the method of proving the Tauberian theorem can be adapted to construct ν1 and ν2 as needed. Note that the
construction given in this paper involves a “local inversion” of eµ1 and µe2 away from their zero sets and thus relates to the problem of division conidered in the pioneering works of Ehrenpreis [12, 13, 14].
Acknowledgement. Part of the research work for this paper was completed during the visits of the three authors at the National Center for Theoretical Sciences in Hsinchu, Taiwan. They would like to thank NCTS (Hsinchu) for partial support of this research. They take great pleasure in expressing their thanks to Professor Winnie Li, the director of NCTS, for the invitation and the warm hospitality during their visits in Taiwan. The second author would also like to use this opportunity to thank the Academia Sinica in Taipei for the invitation during the summer of 2007. He would also like to thank the colleagues at the Sinica for the warm hospitality during his visit in Taipei.
2. Heisenberg Group, Problem of Deconvolution
The Heisenberg group Hn can be given by coordinates [z, t] ∈ Cn× R with group law defined by
[z, t] · [w, s] = [z + w, t + s + 2Im z · ¯w].
This group can be realized as the boundary of the Siegel upper half space Un+1 in Cn+1
∂Un+1 = {(z, zn+1) ∈ Cn+1: Im zn+1 = |z|2},
where the group law gives a group action on the hypersurface. As usual, |z|2 =Pn
k=1|zk| 2.
The left invariant vector fields associated to Hn are given by
Zj = ∂ ∂zj + izj ∂ ∂t,
¯ Zj = ∂ ∂ ¯zj − izj ∂ ∂t, for j = 1, . . . , n, and T = ∂ ∂t.
The only non-zero brackets are given by [Zj, ¯Zj] = −2iT , for j = 1, . . . , n. The subLaplacian
is given by 2 = n X j=1 ¯ ZjZj+ ZjZ¯j.
Harmonic analysis on Hn for radial functions f ∈ L∞
0 (Hn) utilizes the Gelfand
trans-form defined using the bounded U (n)-spherical functions. These can be defined as joint eigenfunctions of 2 and iT and are given by
ψkλ(z, t) = ce2πiλte−2π|λ||z|2L(n−1)k (4π|λ||z|2) for (k, λ) ∈ Z+× R∗,
and
Jρ(z) = c
Jn−1(ρ|z|)
(ρ|z|)n−1 for ρ ∈ R+.
These bounded spherical functions are used to form the Gelfand transform for L1 0(Hn),
defined for f ∈ L10(Hn) as ef (p) for p ∈ H by e f (λ; k) = Z Hn f (z, t)ψλ k(z, t)dm(z, t), and e f (0; ρ) = Z Hn f (z, t)Jρ(z)dm(z, t).
The spectrum is given by the Heisenberg fan H composed of a central Bessel ray Hρ and
infinitely many Laguerre rays Hk,± converging to the central Bessel ray. Denote H, Hρ, and
Hk,± as follows H = Hρ∪ (∪∞k=1Hk,+∪ Hk,−) (2.1) = {(0, ρ) : ρ ≥ 0} ∪∪∞k=1{(λ, 4|λ|(k + n 2) : λ ∈ R ∗} . (2.2)
In application of the Gelfand transform, we will need to define Ψ(n−1)k (x) as Ψ(n−1)k (x) =
Z x
0
e−t/2L(n−1)k (t)tn−1dt. In addition we use the notation jn to represent the function jn(x) =
Jn(x)
xn .
Part of our analysis of this problem will include the perspecive of the Weyl calculus and the group Fourier transform. We first define the position and momentum operators P = (P1, . . . , Pn) and Q = (Q1, . . . , Qn), given by Pju(x) = xju(x) and Qju(x) = 1i∂x∂u
j(x).
The Weyl representation is made of the infinite dimensional representations π±λ = e2πi(±λt±λ
1/2x·P +λ1/2y·Q)
for λ ∈ R+\ {0},
and the one-dimensional representations
that can be attained in the limit as λ → 0. The group Fourier transform on Hn is then given by π±λ(f ) = Z R2n+1 f (x, y, t)π±λ(x, y, t)dxdydt, and π(ξ,η)(f ) = Z R2n+1 f (x, y, t)π(ξ,η)dxdydt.
Both of these can be attained from F2n+1 by the following
π(ξ,η)(f ) = F2n+1(f ) (−η, −ξ, 0) ,
and
π±λ(f ) = F2n+1(f ) ∓λ1/2P, −λ1/2Q, ∓λ .
For additional details on this material, please see [4, 1], where this theory is applied for the Pompeiu transform on spheres in Hn.
In Euclidean space Rn, deconvolution takes place on the side of the Fourier transform,
either in the explicit construction of deconvolvers ν1, . . . , νn satisfying the analytic Bezout
equation
b
µ1νb1+ · · · +µbnνbn ≡ 1,
or in verification of existence of such ν1, . . . , νnthrough verification of the H¨ormander strongly
coprime condition
|µb1(ζ)| + · · · + |µbn(ζ)| ≥ C 1 (1 + |ζ|)Ne
−B|Imζ|
for some C, N, B > 0. Observe that the process of deconvolution takes place on the Fourier transform side; the same will be true for deconvolution on the Heisenberg group, Hn. When the measures µ1, . . . , µn are U (n)-radial, it is possible to consider Gelfand
trans-formsµe1, . . . ,µen, directly related to Fourier-Bessel functionsµb1(ρ), . . . ,µbn(ρ) with the radial variable ρ = p|ζ1|2+ · · · + |ζn|2. In both cases of Section 3, which consider U (n)-radial
spaces, the construction of deconvolvers is completed on the Gelfand transform side. As part of this study, we will also consider the connection to the group Fourier transform through the Weyl calculus in Section 4. We primary focus on two specific cases where the measures are U (n)-radial in this paper and will consider more general cases in a later work.
We do not yet have a version of H¨ormander’s strongly coprime condition for the Gelfand transform on Hn. However, given radial measures µ1, µ2 ∈ L10(Hn), deconvolution is still
possible through explicit construction of ν1, ν2 satisfying
(2.3) µe1(p)eν1(p) +µe2(p)eν2(p) ≡ 1
for every p ∈ H, i.e. for every p = (λ, k) ∈ R∗× Z+ or p = (0, ρ) ∈ R+. Although we cannot
accomplish this in one step for the entire Heisenberg brush H, we do accomplish this in the limit through use of a compact exhaustion {Kj} of H. This is to say we construct sequences
{ν1,j} and {ν2,j} satisfying (2.3) for all p ∈ Kj for each j. The process of deconvolution is
then accomplished in passing to the limit.
Once this process of deconvolution is accomplished, we also consider the relation to the group Fourier transform using the Weyl calculus.
3. Procedure for Deconvolution: Two Specific Cases
Here we consider deconvolution of the Pompeiu transform for the function space C ∩ L∞(Hn) as described in previous papers [1, 4]. It is at this level where two sets are required,
and we first consider the case of a ball and sphere of the same radius in Section 3.1, followed by the case of two balls of appropriate radii in Section 3.2. Although both cases follow the same general procedure, the first is somewhat more direct because we can find a uniform separation of the zeros of the Gelfand transforms for the two sets. In each case we consider a limiting procedure, which allows us to construct a sequence of functions converging to the original function f . To do this we will construct a sequence of deconvolvers {ν1,j} and {ν2,j}
such that
e
T1eν1,j + eT2eν2,j → 1, in the sense that
e
T1νe1,j+ eT2νe2,j ≡ 1
on some compact set Kj, where the sequence {Kj} forms a compact exhaustion of the
Heisenberg fan.
3.1. Ball and Sphere. Here we consider the Pompeiu transform defined in terms of the integral averages Z |z|=r Lgf (z, 0)dσr(z) for all g ∈ Hn, and Z |z|<r Lgf (z, 0)dµr(z) for all g ∈ Hn.
These may also be written as convolution equations f ∗ Sr and f ∗ Tr, where
hφ, Sri = Z |z|=r φ(z, 0)dσr(z) and hφ, Tri = Z |z|<r φ(z, 0)dµr(z).
To define sequences of sets {Kj} and {Vj} we first list the zeros of Jn(rx) seqentially as
{λ1, λ2, . . . , λn, . . .} as well as the zeros of Jn−1(rx), listed sequentially as {λ01, λ02, . . . , λ0n, . . .}.
Then, letting Nj = 3λj+λ0j+1 4 and N + j = λj+λ0j+1 2 , we form Kj = {p = (x, y) ∈ H : x2+ y2 ≤ Nj2, where (x, y) = (λ, |λ|(4k + 2)) or (x, y) = (0, ρ 2)} and Vj = {p = (x, y) ∈ H : x2+ y2 < Nj+ 2 , where (x, y) = (λ, |λ|(4k + 2)) or (x, y) = (0, ρ2)}. We claim the following.
Theorem 1. Let Sr and Tr be the distributions defined above. Consider the sequence of
compact sets {Kj} ⊂ H, which forms a compact exhaustion of the Heisenberg fan H, as
given above. There exist sequences of functions {ν1,j} and {ν2,j} with the property that
e
Srνe1,j+ eTreν2,j ≡ 1 on Kj It is also true that
e
Sreν1,j + eTreν2,j ≡ 0 on V
c j
In the following proof, we furthermore give a method for constructing these sequences of functions {ν1,j} and {ν2,j}. At the end of this section, we illustrate how these sequences are
used in the problem of deconvolution. Proof:
Recall that for f ∈ L1
0(Hn), the Gelfand transform ef is defined on the Heisenberg fan H
which was defined in (2.1). This is realized as a subset of the upper half plane of R2. We
use the compact exhaustion {Kj} given before the statement of this theorem. Note each Kj
is compact, and ∪jKj = H. We will construct the sequences {ν1,j} and {ν2,j} so that ν1,j
and ν2,j satisfy the needed relation
e
Sreν1,j + eTrνe2,j = 1
on Kj, for j = 1, . . . , n. Let us recall the values of eSr and eTr. First,
e Sr(λ, k) = ce−2π|λ|r 2 L(n−1)k (4π|λ|r2), and e Sr(0, ρ) = c Jn−1(rρ) (rρ)n−1 . Likewise e Tr(λ, k) = c Z r 0 e−2π|λ|s2L(n−1)k (4π|λ|s2)s2n−1ds = c0Ψ(n−1)k (4π|λ|r2), and e Tr(0, ρ) = c Jn(rρ) (rρ)n .
We consider the two sets V1 = {zeros of eSr} and V2 = {zeros of eTr}. At times it will be
necessary to focus on the zeros in the Bessel part of the spectrum. For this purpose we define U1 = V1∩ Hρ and U2 = V2∩ Hρ. In order to focus on those zeros of V1 and V2 which
are contained in the set Kj of the compact exhaustion {Kj}, we define the sequences {V1,j}
and {V2,j} by V1,j = V1∩ Kj and V2,j = V2∩ Kj. In order to focus on the zeros inside of Kj
which are in Bessel part of the spectrum, we similarly form the sequences {U1,j} and {U2,j},
defined as U1,j = U1∩ Kj and U2,j = U2∩ Kj.
The next goal is to construct appropriate neighborhoods of the elements of V1 and V2.
These neighborhoods will be used together with “local identities” on each of the neighbor-hoods in the deconvolution procedure. We work outward from the central Bessel ray Hρ,
containing the zeros U1 and U2, since each neighborhood of one of these zeros will contain
an infinite number of zeros in the Laguerre rays. We index the sets of Bessel zeros U1 = {M1, M2, . . . , Mn, . . .},
and
U2 = {N1, N2, . . . , Nn, . . .}.
We know, because of the relation of zeros of Bessel functions of consecutive indices, that these zeros are interlacing, i.e.,
M1 < N1 < M2 < N2 < · · · .
We form sequences of neighborhoods {Ci} and {C
0
i}, where Ci = [(Ni−1+Mi)/2, (Mi+Ni)/2]
is a neighborhood of Mi and C
0
C1, use C1 = [0, (M1 + N1)/2]. We next list some important properties of these sequences.
First, they cover the central Bessel ray, Hρ.
(∪∞i=1Ci) ∪
∪∞i=1Ci0⊃ Hρ.
Next, they are nearly disjoint, with consecutive neighborhoods intersecting only at the end-points: Ci∩ C 0 i = {(Mi+ Ni)/2} and C 0 i ∩ Ci+1= {(Ni+ Mi+1)/2}.
Finally, they separate the zero sets of eSr and eTr. In particular, for all j,
Nj∩ (∪∞i=1Ci) = ∅ and Mj ∩
∪∞i=1Ci0= ∅. Letting δi = (Ni − Mi)/2 and δ
0
i = (Mi+1− Ni)/2, where for δ
0
0 we use N0 = 0, we also
form the smaller neighborhoods Bi = [Mi− δ
0
i−1/2, Mi+ δi/2] of Mi,
and similarly Bi0 = [Ni− δi/2, Ni+ δ
0
i/2] of Ni.
This gives sequences of neighborhoods {Bi} and {B
0 i}, satisfying Bi ⊂ Ci and B 0 i ⊂ C 0 i.
We furthermore claim there exist sequences of slightly larger neighborhoods {Vi} and {V
0
i}
where Vi satisfy Ci ⊂ Vi and Vi ∩ B
0
i−1∪ B
0
i
= ∅. Likewise Vi0 satisfy Ci0 ⊂ Vi0 and Vi0∩ (Bi−1∪ Bi) = ∅.
The next step is to extend the neighborhoods from the Bessel ray Hρ to all of the
Heisenberg fan H. Recall that as k → ∞, the Laguerre rays Hk,± converge to Hρ. For
any given i, we expand the neighborhood Bi in Hρ to a wider neighborhhod Bi,j in the
Heisenberg fan H such that Bi,j ∩ Hρ = Bi. Note that in H, {Bi} can be written as
{{0} × [Mi− δ−i /2, Mi+ δi+/2]} and that the Laguerre rays Hk,± can be expressed as
Hk,± = {(λ, 4|λ|(k + n/2) : λ ∈ R∗)}.
We then define Bi,j by
Bi,j = {(x, y) : Mi− δ−i /2 2 ≤ x2+ y2 ≤ M i+ δi+/2 2 and |y x| ≥ 4(j + n/2)}. We then want to choose ji ∈ Z+ with the property that, for each j ≥ ji exactly one of the
Laguerre zeros on the ray Hj,± is inside of Bi,j ∩ Hj,±. We will also choose ji to minimize
all possible ji satisfying this property, and consider the neighborhood Bi,ji. Using the same
choice for ji, we also consider the larger neighborhoods Ci,ji and Vi,ji. These yield sequences
of neighborhoods {Bi,ji}, {Ci,ji}, and {Vi,ji}. We use the identical construction for sequences
of neighborhoods {B0 i,ji0}, {C 0 i,ji0}, and {V 0 i,j0i}.
Considering the sequences {Ci,ji} and {C 0
i,ji0}, it remains true that
Hρ ⊂ (∪∞i=1Ci,ji) ∪ ∪∞i=1Ci,j0 0 i .
However, in general, the above union will not cover all of the Heisenberg fan H, and further-more, there will be some of the zeros in the sets V1 and V2 which are not covered. However
there can only be a finite number of such zeros on each Laguerre ray. Furthermore these remaining zeros will be locally finite. In order to cover these zeros, we first denote them as
{P1, P2, . . . , Pn, . . .} and {P 0 1, P 0 2, . . . , P 0 n, . . .}.
There exist neighborhoods {D1, D2, . . . , Dn, . . . } and {D 0 1, D 0 2. . . , D 0 n, . . .},
as well as neighborhoods {W1, W2. . . , Wn, . . .} and {W
0 1, W 0 2, . . . , W 0 n, . . .} satisfying: (∪∞i=1Di) ∪
∪∞i=1Di0⊃ H \h(∪∞i=1Ci,ji) ∪
∪∞i=1Ci,j0 0 i i where 1.Di ⊂ Wi and D 0 i ⊂ W 0 i 2.Pi∩ ∪∞i=1W 0 i = ∅ and P 0 i ∩ (∪∞i=1Wi) = ∅ 3.Pi 6∈ Wj for i 6= j and P 0 i 6∈ W 0 j for i 6= j.
We have found neighborhoods {Ci,ji} and {Di}
∞
i=1 of V1, the zeros of eSr, as well as
neigh-borhoods {C0
i,j0i} and {D
0
i} ∞
i=1 of V2, the zeros of eTr, such that
H ⊂ (∪Ci,ji) ∪ (∪ ∞ i=1Di) ∪ ∪C0 i,ji0 ∪∪∞i=1Di0, and such that there is separation between V1 and
∪C0 i,ji0 ∪ ∪∞ i=1D 0 i, as well as between V2 and (∪Ci,ji) ∪ (∪ ∞
i=1Di). We next find sequences of “local identities” which equal 1
on these neighborhoods. These are used to “invert” eSr on neighborhoods of V2 and eTr on
neighborhoods of V1, away from their zero sets. These are combined to “invert” the transform
on all of H.
We use the result on “local identities” that for K a compact subset of H and F and open subset of H such that K ∩ F = ∅, there exists ρ ∈ L10(Hn) such that ρ|eK ≡ 1 and ρ|eF ≡ 0. Also require that 0 ≤ ρ ≤ 1. We thus find the sequences of functions {ρe 1,i} and {π1,i}∞i=1
which satisfy e ρ1,i = (1 on C0 i,ji0 0 on F0 i,j0i, where F0 i,j0i = V0 i,ji0 c . And, in addition, e π1,i = ( 1 on D0i 0 on (Wi0)c. Similarly we will need the sequences {ρ0,i} and {π0,i}ti=1 satisfying
e ρ0,i =
(
1 on Ci,ji
0 on Fi,ji,
where Fi,ji = (Vi,ji)
c . And, in addition, e π0,i = ( 1 on Di 0 on (Wi)c.
We begin with ρe0,1, which is identically 1 on C1,j1, a set which includes the neighborhood
along the Bessel ray Hρ containing the origin, i.e. (0, [0, (M1 + N1)/2]) ⊂ Hρ. Next, we
extend to a function whose transform is identically 1 on C1,j1 ∪ C 0
ρ1 = ρ1,j + ρ0,j − ρ1,j∗ ρ0,j. Then e ρ1 =ρe1,1+ρe0,1−ρe1,1ρe0,1 = (1 on C1,j1 ∪ C 0 1,j10 0 on F1,j1 ∩ F 0 1,j01.
The above procedure demonstrates how “local identities” for the sets C1,j1 and C 0
1,j10 are
joined to form a “local identity” on their union. Note also that these cover the Bessel zeros, as well as an infinite number of the Laguerre zeros. Thus only a finite number of Laguerre zeros are left, and these can be covered by neighborhoods {D1, . . . , Ds1} and {D
0
1, . . . , D
0
1,t1}.
Next expand to form a “local identity” for K1by adjoining the “local identities” π0,1, . . . , π0,s1
of D1, . . . , Ds1 and π1,1, . . . , π1,t1 of D 0 1, . . . , D 0 t1, respectively. Then π 1 0 = Ps1 i=1π0,i is an identity for D1 = ∪s1 i=1Di, while π11 = Pt1
i=1π1,i is an identity for D 10 = ∪t1 i=1D 0 i. First, let π1 = π1
0+ π11− π10∗ π11, and note this will satisfy
e π1 =eπ01+πe11−eπ01eπ11 = ( 1 on D1∪ D10 0 on (W1)c∩ (W10 )c, where (W1)c = ∩s1 i=1(Wi)c and (W1 0 )c = ∩t1 i=1(W 0 i)c. Noting that K1 ⊂ C1,j1 ∪ C 0 1,j01 ∪ D1∪ D10, to expand the “identity” to K
1 by forming ρ1∗= ρ1+ π1− ρ1∗ π1. The Gelfand
transform of ρ1∗ will satisfy
e ρ1∗ =ρe1+eπ1−ρe1eπ1 = 1 on C1,j1 ∪ C 0 1,j10 ∪ D1∪ D10 0 on (F1,j1 ∩ F 0 1,j01) ∩ ((W 1)c∩ (W10 )c),
thus forming an “identity” for K1. Finally note that ρ1∗1 = ρ1,1 + π11 is an “identity” on
C0
1,j10
∪ D10, a neighborhood of all the zeros of eS
r in K1, that is also separated from the zeros
of eTr. We can form ρ1∗2 = ρ1∗− ρ1∗1 , and note this function will similarly give an “identity”
on A1 = (C1,j1 ∪ D
1) \V0
1,j10 ∪ D 10
, a neighborhood of all zeros of eTr in K1, also separated
from the zeros of eSr.
This construction is next extended to form “identities” for the Kj in the compact
exhaus-tion of H. Form neighborhoods Cm = ∪mi=1Ci,ji and C 0
m = ∪mi=1C
0
i,ji0 for the zeros of eTr and
e
Sr, respectively, inside of Km and near the Bessel ray Hρ. Note that “identities” for Cm and
Cm0 are given by ρm0 = Pm
i=1ρ0,i and ρm1 =
Pm
i=1ρ1,i. Letting ρm = ρm0 + ρm1 − ρm0 ∗ ρm1 , we
have that e ρm =ρem0 +ρe1m−ρem0 ρem1 = ( 1 on Cm∪ Cm0 0 on Fm∩ Fm0, where Fm = ∩m
i=1Fi,ji and F
m0 = ∩m i=1F
0
i,ji0. Then let D
m = ∪sm i=1Di and Dm 0 = ∪tm i=1D 0 i. We also define (Wm)c = ∩sm i=1(Wi)c and (Wm 0 )c = ∩tm i=1(W 0
i)c. Expanding to pick up the
remaining Laguerre zeros in Km, we let πm0 =
Psm i=1π0,i π m 1 = Ptm i=1π1,i π m = πm 0 + π1m − πm 0 ∗ π1m e πm =eπm0 +eπ1m−eπ0meπ1m = ( 1 on Dm∪ Dm0 0 on (Wm)c∩ (Wm0)c.
Finally, we can expand to all of Km by forming ρm∗= ρm+ πm− ρm∗ πm where e ρm∗ =ρem+eπm−ρemeπm = ( 1 on Cm∪ C 0 m∪ (Dm∪ Dm 0 ) 0 on (Fm∩ Fm0) ∩ ((Wm)c∩ (Wm0)c). Noting that Km ⊂ Cm∪ C 0 m ∪ Dm∪ Dm 0
, we see that ρm∗forms an “identity” for K m. As
above, we split ρm∗into two “identities” ρm∗
1 and ρm∗2 which are “identities” on neighborhoods
of the zeros of eSr and eTr, respectively. First ρ1m∗= ρm1 + π1m is an “identity” on Cm
0
∪ Dm0
, a neighborhood of all the zeros of eSr in Km, that is also separated from the zeros of eTr.
We can form ρm∗
2 = ρm∗− ρm∗1 , and note this function will similarly give an “identity” on
Am = (Cm∪ Dm) \ Vm0∪ Dm0, a neighborhood of all zeros of
e
Tr in Km, also separated
from the zeros of eSr.
We next form the deconvolving sequences {ν1,j} and {ν2,j}. Here we will rely on the
separation of the zeros of eSr from those of eTr, and we utilize the neighborhoods and“local
identities” formed above in the process of inversion. In particular, eSrdoes not vanish on ¯V
0 = ∪∞ i=1V 0 i,j0i ∪ ∪∞ i=1D 0
i, and similarly Ter does not vanish on ¯V = (∪∞i=1Vi,ji) ∪ (∪
∞
i=1Di). We
first invert these on the set Kj, and for this purpose, we form ¯V
0 j = ¯V 0 ∩ Kj and ¯Vj = ¯V ∩ Kj. Let Mj,s = minx∈ ¯V0 j Ser(x) and Mj,t = minx∈ ¯Vj Ter(x)
. Then let Mj = min{Mj,s, Mj,t}. There exists φj ∈ C∞ such that φj(t) = 1t for |t| ≥ Mj, while φ(t) = 0 for |t| ≤ Mj/2. Then
form φj◦ Sr and φj◦ Tr, and we rely on the lemma [4, Lemma 6.4.4] to describe the Gelfand
transforms (φj ◦ Sr)e and (φj ◦ Tr)e. Notice these invert eSr on the set ¯V
0
j and eTr on the set
¯
Vj, away from their zero sets, as follows
(φj◦ Sr)e|V¯0 j = φj e Sr |V¯0 j = 1/ eSr, and (φj ◦ Tr)e|V¯j = φj e Tr |V¯j = 1/ eTr.
The deconvolving sequences {ν1,j} and {ν2,j} are now formed by ν1,j = ρ1,j ∗ (φj◦ Sr) and
ν2,j = ρ2,j∗(φj◦ Tr). We claim these sequences satisfy the properties claimed in the theorem.
Considering {Sr∗ ν1,j + Tr∗ ν2,j}, we form the Gelfand transforms
(Sr∗ ν1,j+ Tr∗ ν2,j)e = Sereν1,j+ eTreν2,j = Serφj e Sr e ρ1,j + eTrφj e Tr e ρ2,j.
Since φj( eSr)|supp(eρ1,j)∩Kj ≡ 1/ eSr and φ( eTr)|supp(eρ2,j)∩Kj ≡ 1/ eTr, we have
(Sr∗ ν1,j+ Tr∗ ν2,j)e|Kj = ρe1,j|Kj· eSr· 1/ eSr +ρe2,j|Kj · eTr· 1/ eTr = ρe1,j|Kj+ρe2,j|Kj.
Thus by design of {ρ1,j} and {ρ2,j}, we have that
(Sr∗ ν1,j+ Tr∗ ν2,j)e|Kj = (ρe1,j+ρe2,j) |Kj ≡ 1,
and
(Sr∗ ν1,j + Tr∗ ν2,j)e|(Uj)c = (ρe1,j+ρe2,j) |(Uj)c ≡ 0.
This case yields a nice representative example on which we have demonstrated the more general method. Although there was a uniform separation of zeros of eSr and eTr in the above
case, more generally, these zero sets may coalesce. In the case of two balls of appropriate radii considered in the next example, the zeros of eTr1 and eTr2 will coalesce. However, we will
be able to extend the above method by taking enough care with the neighborhoods around these coalescing zeros.
3.2. Two Balls of Appropriate Radii. We now consider the Pompeiu transform defined in terms of the integral averages
Z |z|<r1 Lgf (z, 0)dµr1(z) for all g ∈ Hn, and Z |z|<r2 Lgf (z, 0)dµr2(z) for all g ∈ Hn.
These may also be written as convolutions f ∗T1and f ∗T2, where hφ, T1i =
R
|z|<r1φ(z, 0)dµr(z)
and hφ, T2i =
R
|z|<r2φ(z, 0)dµr(z). We consider the case where r1 and r2 are such that the
transform is injective. We claim the following.
Theorem 2. We assume that r1 and r2 satisfy the conditions
1. r1 r2 6∈ Q(Jn) = {γxγy : Jn(x) = Jn(y) = 0, γ ∈ R∗}, 2. r1 r2 2
6∈ Q(Ψ(n−1)k ) = {γxγy : Ψk(n−1)(x) = Ψ(n−1)k (y) = 0, γ ∈ R∗}, for all k ∈ Z+.
Then eT1 and eT2 do not have any common zeros. Consider the sequence of compact sets
{Kj} ⊂ H given below, which forms a compact exhaustion of the Heisenberg fan H. There
exist sequences of functions {ν1,j} and {ν2,j} with the property that
e T1νe1,j + eT2eν2,j ≡ 1 on Kj, and e T1eν1,j+ eT2νe2,j ≡ 0 on V c j,
where each Vj is an open set such that Kj ⊂ Vj ⊂ Kj+1.
Recall that Ψ(n−1)k (x) was defined above by Ψ(n−1)k (x) =
Z x
0
e−t/2L(n−1)k (t)tn−1dt.
In this case the {Kj} and {Vj} will be given in the proof of the theorem. It is more convenient
to make their definition after the zeros have been grouped appropriately. Proof: We first recall the values of the Gelfand transforms of T1 and T2
e T1(λ, k) = c Z r1 0 e−2π|λ|s2s2n−1L(n−1)k (4π|λ|s2)ds = c0Ψ(n−1)k (4π|λ|r12) e T1(0, ρ) = c Jn(ρr1) (ρr1)n ,
and e T2(λ, k) = c Z r2 0 e−2π|λ|s2s2n−1L(n−1)k (4π|λ|s2)ds = c0Ψ(n−1)k (4π|λ|r22) e T2(0, ρ) = c Jn(ρr2) (ρr2)n .
The procedure is close to that of the previous theorem; however there are some additional complications we will address. The interaction between zero sets of eT1 and eT2 plays a larger
role. In general the zeros will not alternate, and furthermore there is not a uniform separation between the zeros of eT1 and the zeros of eT2. Due to the coalescing zeros of these sets, the
values of eT1 can be very small near the zeros of eT2. However, by being careful with the size
of the sets enclosing the zeros, we can still solve this problem of constructing the appropriate sequences {ν1,j} and {ν2,j}. Nevertheless, the size of eT1 near the zeros of eT2 can still be an
issue in the larger problem of deconvolution, as addressed in Section 5.
The procedure is just like the above case, and we begin by forming appropriate neigh-borhoods of the zeros of eT1 and eT2 along the Bessel ray. Letting V1 = {zeros of eT1} and
V2 = {zeros of eT2}, we have the Bessel zeros U1 = V1 ∩ Hρ and U2 = V2 ∩ Hρ. Letting
U1 = {M1, M2, . . .} and U2 = {N1, N2, . . .}, we let U = U1 ∪ U2 = {Z1, Z2, . . .} where, in
each case, these are listed in increasing order. We next form sequences of neighborhoods {Ck} and {C 0 k} such that ∪ ∞ k=1 C 0 k∪ Ck = Hρ, and so that ∪∞k=1C 0 k covers U1, while ∪∞k=1Ck
covers U2. Beginning with the first zero Z1 = M1 ∈ U1, we find the next Zj1 equal to
N1 ∈ U2. Then Z1, . . . , Zj1−1 are grouped as zeros of eT1, and C1 = [0,
Zj1−1+Zj1
2 ] is a
neigh-borhood of these zeros of eT1 that does not contain any zeros of eT2. Since Zj1 ∈ U2 and
we know Zj1+1 = Mj1 ∈ U1, we also form C 0 1 = [ Zj1−1+Zj1 2 , Zj1+Zj1+1 2 ] as a neighborhood of
N1 ∈ U2 not containing any zeros of eT1. Using Zj1+1 = Mj1, we then find the next Zj2 that
equals N2 ∈ U2. Then the zeros Zj1+1, . . . , Zj2−1 are grouped as zeros of eT1 not separated
by any zeros of eT2, and we may form a neighborhood C2 = [
Zj1+Zj1+1 2 , Zj2−1+Zj2 2 ]. Then let C20 = [Zj2−1+Zj2 2 , Zj2+Zj2+1
2 ] to give a neighborhood of N2 = Zj2. Extending this procedure to
all integers yields the desired collection of neighborhoods {Ck} covering U1 and {C
0
k}
cov-ering U2 such that Hρ= ∪∞k=1 Ck∪ C
0
k. Similarly to the above proof these neighborhoods
also separate the zero sets of eT1 and eT2. In particular, for all j,
Nj∩ (∪∞i=1Ci) = ∅ and Mj∩
∪∞i=1Ci0= ∅.
These collections of neighborhoods are also nearly disjoint, intersecting only at the endpoints, as above. This point in the discussion will also be convenient to define the sequences of sets {Kj} and {Vj}. Let Ni = 3Zji+Zji+1 4 and N + i = Zji+Zji+1 2 . We then let Kj = {p = (x, y) ∈ H : x2+ y2 ≤ Nj2, where (x, y) = (λ, |λ|(4k + 2)) or (x, y) = (0, ρ 2)} and Vj = {p = (x, y) ∈ H : x2+ y2 < Nj+ 2 , where (x, y) = (λ, |λ|(4k + 2)) or (x, y) = (0, ρ2)}, as above.
For an additional distance in the separation, we also form smaller neighborhoods {Bi}
and {Bi0}, satisfying Bi ⊂ Ci and B
0
i ⊂ C
0
zeros {Zji−1+1, . . . , Zji−1} ∈ U1, where j0 = 0, while B 0
i is a neighborhood of the zeros
Zji ∈ U2. This can be accomplished by letting δi =
Zji−Zji−1 2 and δ 0 i = Zji+1−Zji 2 . Then form Bi = [Zji−1+1−δ 0 i−1/2, Zji−1+δi/2] and B 0 i = [Zji−δi/2, Zji+δ 0
i/2]. These neighborhoods have
the desired properties, and they further guarantee that dist(U1, B
0 i) = iand dist(U2, Bi) = 0 i, where i = min(δi/2, δ 0 i/2) and 0 i = min(δi/2, δ 0
i−1/2), giving a local separation of zeros.
Although this local separation of zeros is not as strong as the uniform separation of zeros we find above in Section 3.1, it is good enough for the purpose of forming the desired deconvolving sequences. In particular, the distance in the local separation of the zeros allows us to extend these neighborhoods beyond the Bessel ray into the Heisenberg fan, as above. But first we observe the existence of the collections of larger neighborhoods {Vi}
and {Vi0}. The Vi satisfy the properties Ci ⊂ Vi and Vi ∩ B
0
i−1∪ B
0
i = ∅. Likewise, the
Vi0 satisfy Ci0 ⊂ Vi0 and Vi0 ∩ (Bi−1∪ Bi) = ∅. Using the collection of neighborhoods {Bi},
{Ci}, and {Vi} of the zeros U1 ∈ Hρ and the collection of neighborhoods {B
0
i}, {C
0
i}, and
{Vi0} of the zeros U2 ∈ Hρ, it is possible to complete the process of forming the deconvolving
sequence by a sequence of steps beginning with extension from the Bessel ray, Hρ. The key
point in making the extension from the Bessel ray is the local separation by a set i between
the zeros U1 and neighborhoods B
0
i, as well as between the zeros U2 and neighborhoods Bi.
We note that it is straightforward to see how the neighborhoods for this specific case could be extended to a more general case in which the zero sets are more irregularly distributed. Once the appropriate neighborhoods have been established, the remainder of the formation of the deconvolting sequences proceeds directly as described.
From here we complete the process of forming the deconvolving sequences using the same method as given in Section 3.1 for Theorem 3.1. It consists of extension of the above neighborhoods from the Bessel ray to cover an infinite number of Laguerre zeros, followed by formation of additional neighborhoods to cover the remaining Laguerre zeros and any remaining regions in the Heisenberg fan H. From here, inversion of eT1 and eT2 are completed
by use of “local identities” and a process of “local inversion”. The local separation between the zeros U1, U2 and the neighborhoods Bi and Bi0 described in the preceding paragraph
extends to the entire Heisenberg fan H. This is used to form the φi which locally invert eT1
and eT2. In addition the system of neighborhoods are used to form the sequences {ρ1,j} and
{ρ2,j} with the property thatρe1,j+ρe2,j
Kj ≡ 1. Putting these together will yield the desired
deconvolving sequences {ν1,j} and {ν2,j}.
The method for extending the neighborhoods {Bi,ki} and {B
0
i,k0i}, as well as {Ci,ki} and
{C0 i,k0
i} plus {Vi,ki} and {V
0 i,k0
i} is identical to that given above for Theorem 2.1. We outline
this procedure again here. Recall the definition of Bi,j, now expanded to enclose the group
of zeros Zji−1+1, . . . , Zji−1 Bi,k = n (x, y) ∈ H : Zji−1+1− δ0i−1 2 2 ≤ x2+ y2 ≤Zji−1+ δi 2 2 and y x ≥ 4(k + n/2) o , while Bi,k0 0 is defined by
Bi,k0 0 = n (x, y) ∈ H :Zji− δi 2 2 ≤ x2+ y2 ≤Z ji+ δ0i 2 2 and y x ≥ 4(k 0 + n/2)o. The number of Bessel zeros inside of Bi is expanded to Bi,ki by choosing ki ∈ Z+ with
the property that, for each k ≥ ki, exactly ni of Laguerre zeros on the ray Hk,± is inside of
use this to form the neighborhood Bi,ki. Similarly, B
0
i is expanded to form Bi,k0 0
i by choosing
k0i ∈ Z+ with the property that, for each k0 ≥ ki0, exactly one of the Laguerre zeros on the
ray Hk0,± is inside of B0
i,k0i∩ Hk0,±. Using these same values of ki and k
0
i, for each i ∈ Z+, we
also expand from {Ci} to {Ci,ki} and from {C 0
i} to {C 0
i,k0i}. In the same manner we expand
from {Vi} to {Vi,ki} and from {V 0
i} to {Vi,k0 0
i}. Finally pick up remainder of Laguerre zeros
and cover the rest of H by adding neighborhoods {Di} and {D0i} as well as neighborhoods
{Wi} and {Wi0} that separate the previous neighborhoods, using the same method as given
in the proof of Theorem 3.1.
Now that all the neighborhood systems are in place, the formation of the deconvolving sequences is identical to that given in Theorem 3.1. Sequences of “local identities” {ρ1,i}
and {ρ0,i} are formed, such that
e ρ1,i = ( 1 on Ci,j0 0 i 0 on Fi,j0 0 i, and e ρ0,i = ( 1 on Ci,ji 0 on Fi,ji.
Similarly we form sequences of “local identities” {π1,i} and {π0,i} such that
e π1,i = ( 1 on D0i 0 on (Wi0)c, and e π0,i = ( 1 on Di 0 on (Wi)c.
These are put together in the same manner as in the proof of Theorem 3.1, forming ρ∗m which is an “identity” for Km. This further splits into the two “identities” ρm∗1 and ρm∗2 such
that ρ∗m = ρm∗
1 + ρm∗2 , while ρm∗1 and ρm∗2 are “identities” on the neighborhoods of the zeros
of eT1 and eT2, respectively.
All that remains is the “local inversion” of eT1 and eT2 away from their zeros, found by
applying the inverses φj ∈ C∞ satisfying φj(t) = 1/t for |t| ≥ Mj while φj(t) = 0 for
|t| ≤ Mj/2, where the Mj are determined according to the size of eT1and eT2on the appropriate
neighborhood systems within Kj. Note that eT1 does not vanish on ¯V0 =
∪∞ i=1Vi,j0 0 i ∪ (cup∞i=1D0i), and similarly eT2 does not vanish on ¯V = (∪∞i=1Vi,ji) ∪ (∪
∞ i=1D
0
i). For inversion on
Kj, we divide into neighborhoods of the zero sets ¯Vj0 = Kj ∩ ¯V0 and ¯Vj = Kj ∩ ¯V . Then
Mj = min{Mj,1, Mj,2}, where Mj,1 = minx∈ ¯V0 j Te1(x) and Mj,2 = minx∈ ¯Vj Te2(x) . Note that (φj◦ T1) and (φj ◦ T2) invert eT1 and eT2 on ¯Vj0 and ¯Vj, respectively, away from their zero sets,
as follows
(φj◦ T1)e|V¯j0 = φj( eT1)|V¯j0 = 1/ eT1, and
Note that the main difference between this case and that of Theorem 3.1 is the existence of coalescing zeros in this case. To deal with the locations these zeros coalesce, we have more variation in the width of the neighborhoods Bi0, extending to the neighborhoods Bi,j0 0
i,
which separate the zeros of eT2 from those of eT1. The neighborhood systems hold up so that
the deconvolving sequences can be formed. However, the rate at which the zeros coalesce can effect the rate at which Mj approaches 0 as well as the corresponding rate at which
the (φj ◦ T1)e grows near the zero set of eT2. These issues and other related issues will be considered in Section 5 dealing with issues of convergence related to the sequences of deconvolvers.
The deconvolving sequences {ν1,j} and {ν2,j} are now formed by defining ν1,j = ρ1,j ∗
(φj◦ T1) and ν2,j = ρ2,j ∗ (φj◦ T2). As previously, this gives the transforms
(T1∗ ν1,j + T2∗ ν2,j)e|Kj = ρe1,j|Kj· eT1· 1/ eT1+ρe2,j|Kj · eT2· 1/ eT2
= ρe1,j|Kj+ρe2,j|Kj ≡ 1.
We can also easily see that (T1∗ ν1,j + T2∗ ν2,j)e|(Vj)c = 0. This verifies the claims in the
theorem, and the proof is complete.
As noted above, the difference in these cases of the sphere and ball in Section 3.1 and the two balls of appropriate radii in Section 3.2 is the issue of distance of separation of the zeros for eµ1 and µe2, which carries over to the rates of growth of the sequences {eν1,j} and {νe2,j} near these zero sets as j becomes infinite. This issue and related issues of convergence
will be revisited in Section 5, where we also address how the deconvolving sequences {νe1,j}
and {eν2,j} may be applied to recover the function f .
4. Using Weyl Calculus and the Group Fourier Transform
Observe, how in the proof of the results Theorem 3.1 and Theorem 3.2 above, the construction of the “local inverses” ρ1 and ρ2 was based on analysis of the Bessel zeros, sets
U1 and U2. The neighborhoods of each of these zeros naturally extended to Laguerre zeros
along an infinite number of Laguerre rays, based on the subspace topology. The remaining Laguerre zeros are finite along each of the Laguerre rays and are also locally finite. Since these zeros are easily incorporated using a finite number of appropriate neighborhoods, they do not affect the process. Thus, the construction of the “local inverses” in Section 3 above is based upon the construction for the Bessel ray. It appears that the distribution of the zeros along the Bessel ray determine the potential for deconvolution, provided there are no common Laguerre zeros. In the context of the Weyl calculus, the Laguerre spectrum is a quantization of the Bessel part, and the behavior of the Laguerre part, in the limit as k → ∞ and λ → 0, determines the behaviour of the Bessel part.
Recall the suggestion [10] that use of the Weyl calculus can establish connections between Pompeiu type results in Euclidean space and the Heisenberg group. Furthermore in [11] there was also the suggestion that the cases of the Pompeiu problem for the Heisenberg group and Euclidean space are actually very close, and in particular deconvolution should extend to the Heisenberg case. In this section we investigate specific points regarding how the deconvolution results of Theorem 3.1 and Theorem 3.2 can be viewed primarily from the perspective of the “Euclidean part” of the zero set, along the Bessel ray.
In this section we discuss the relations between these two avenues of investigation are closely related. The Weyl representation reduces to this central Bessel ray for the represen-tations π(ξ,η). The issue of extending the deconvolution process from the Bessel ray Hρ to
the Laguerre rays ∪k∈Z+Hk,± has parallels to the passage between the operator-valued Weyl
representation π±λ on the entire spectrum and the Euclidean part π(ξ,η) on the central Bessel
ray. We focus on the operator-valued Weyl representations π±λ and π(ξ,η) of the
distribu-tions T1 and T2, representing the sets over which the average is taken. Through use of the
Weyl calculus we relate eTj(0; ρ), the Bessel part of the Gelfand transform, to π(ξ,η)Tj, the
Euclidean part of the Weyl representation. This corresponds to analysis of the Bessel zeros Uj for j = 1, 2, mentioned above. In fact the conclusion of the existence of deconvolving
sequences in Theorem 3.1 and Theorem 3.2 can be based on two points, the behavior of the Bessel zeros and non-overlapping of the zeros of the Laguerre part.
The Weyl calculus for the Pompeiu problem on Hn allows a unification of both Laguerre
and Bessel parts of the zero sets U1 and U2 into the kernels of two operator-valued functions.
Furthermore, in the case where λ → 0, this carries over to the representation π(ξ,η), the
Euclidean transform on Cn. In this limit, the operator-valued functions become identical to
functions used in the analysis of the Pompeiu problem on Cn. This is a nice bridge from
Heisenberg to Euclidean and from Euclidean to Heisenberg, and we will utilize it. In the case of Theorem 3.1, there are no conditions needed for the common radius of the sphere and ball. In fact, these two sets were selected to provide a representative example for the nice cases where there exists a uniform separation between the zero sets. As such, these sets are in a category comparable to the moment type results of [3, 9, 18] and do not require sets of exceptional radii. This corresponds to the fact that
ker{jn−1(rp|λ|(P2+ Q2))} ∩ ker{jn(rp|λ|(P2+ Q2))} = {0},
a fact which is true for any radius, r. However, in many cases exceptional radii are required, such as Theorem 3.2, corresponding to results such as those in [1, 4, 10]. In this context, we have the two radius theorem for the Pompeiu problem on Hn expressed as follows.
Theorem 3. Let r1 and r2 be two radii satisfying the condition
ker{jn−1(r1p|λ|(P2+ Q2))} ∩ ker{jn−1(r2p|λ|(P2+ Q2))} = {0},
so that eT1 and eT2 do not have any common zeros. Then there exist sequences of functions
{ν1,j} and {ν2,j} with the property that
e T1eν1,j+ eT2eν2,j ≡ 1 on Kj and e T1eν1,j+ eT2νe2,j ≡ 0 on V c j,
where each Vj is an open set such that Kj ⊂ Vj ⊂ Kj+1.
Note that in the case of the n-ball and (n − 1)-sphere of Theorem 3.1, the condition ker{jn(rp|λ|(P2+ Q2))} ∩ ker{jn−1(rp|λ|(P2+ Q2))} = {0}
is automatic due to results of zeros of Bessel functions of consecutive indices and does not relate to the radius r. Similarly there is no condition for exceptional radii needed in Theorem 3.1.
Here we describe the transforms associated to the measures used in Theorem 3.1 and Theorem 3.2 using the operator-valued Fourier transform on Hn. As described in Section
2, the group Fourier transform for the measure µ with respect to the representation π±λ can
be determined from the standard Euclidean Fourier transform on R2n+1 F2n+1(µ)(x, y, t) by
substituting the operators P and Q to give
π±λ(µr) = F2n+1(µr)(∓λ1/2P, −λ1/2, ∓λ) for λ ∈ R+\ {0}.
The case of the one-dimensional measures π(ξ,η), corresponding directly to the Euclidean
case, can be attained as a limit as λ → 0 or by substitution of (ξ, η) in the form π(ξ,η)(µr) = F2n+1(µr)(−ξ, −η, 0) for (ξ, η) ∈ Rn× Rn.
Since the issue of deconvolution on the associated Euclidean spaces is already settled, we may look here first. We first address the issue of common zeros required for injectivity of the Pompeiu transform and inherent to H¨ormander’s strongly coprime condition. In the case of the sphere and ball of Theorem 3.1, we have
π(ξ,η)(Sr) = c Jn−1(rpξ12+ · · · + ξn2+ η21+ · · · + ηn2) (rpξ2 1 + · · · + ξn2 + η12+ · · · + η2n)n−1 , and π(ξ,η)(Tr) = c Jn(rpξ12+ · · · + ξn2+ η21+ · · · + ηn2) (rpξ2 1 + · · · + ξn2+ η12+ · · · + ηn2)n = c0 Z r 0 Jn−1(ρpξ12+ · · · + ξn2+ η21+ · · · + ηn2) (ρpξ2 1 + · · · + ξn2 + η12+ · · · + η2n)n−1 ρ2n−1dρ.
These have no common zeros as a consequence of the well known result for Bessel functions of separate integer indices. In fact, there is a uniform separation among the zeros, which are interlaced. Note that in moving to the infinite dimensional representations and using the series [15] jn−1(ts) = Jn−1(ts) (ts)n−1 = 2 ∞ X j=0 (−1)je−t2/2L(n−1)j (t2)e−s2/2L(n−1)j (s2)
to express the Laguerre part of the spectrum for these operators, we may write π±λ(µr) in
the form π±λ(Sr) = c ∞ X j=0 Fλ(2j + 1)(−1)je−|λ|(P 2+Q2) L(n−1)j 2|λ|(P2+ Q2) ,
which can be represented using H = P2 + Q2, the harmonic oscillator Hamiltonian. Thus
we have π±λ(Sr) = c ∞ X j=0 Fλ(2j + 1)(−1)je−|λ|HL (n−1) j (2|λ|H) .
Noting that H has the Hermite functions Eα as eigenfunctions with eigenvalues 2α + 1,
we see that Ej is also an eigenfunction for the operator e−HL (n−1)
j (2H). This implies that
each (λ, k) ∈ R∗ × Z+ such that L(n−1)k (|λ|r2/2) = 0 yields a function Ek in the kernel of
π±λ(Sr). The kernel of the operator-valued function jn−1(rp|λ|(P2+ Q2)) is then given
by (ξ, η) ∈ Rn× Rn such that J
n−1(rp|ξ|2 + |η|2) = 0 and (λ, k) ∈ R∗ × Z+ such that
separation, due to the indices. To consider the other part of the common kernel, we consider also π±λ(Tr), similarly computed through the use of (4) to be
π±λ(Tr) = c ∞ X j=0 (−1)j Z r 0 e−|λ|ρ2/4L(n−1)j (|λ|ρ2/2)ρ2n−1dρ e−(P2+Q2)L(n−1)j (2(P2+ Q2)), which similarly has the Hermite functions Eα as eigenfunctions. In this case the operator
π±λ(Tr) has a the eigenfunction Ek in its kernel for any (λ, k) ∈ R∗× Z+ such that
Z r 0
e−|λ|ρ2/4L(n−1)k (|λ|ρ2/2)ρ2n−1dρ = 0.
Then recognize that R0r2|λ|/2e−x/2xnL(n−1)j (x)dx and e−r2|λ|/4L(n−1)j (|λ|r2/2) will also have a uniform separation between their zero sets, as was observed previously, in Section 3.1. Thus in this case the entire issue of injectivity, required as a prerequisite for deconvolution, is automatic. Furthermore the uniform separation within the Bessel part of the transform extends to the whole spectrum H. This uniform separation makes the larger problem of deconvolution easier, and this case allows for the most direct application of these methods. We will have more to say later about the role of the zero sets in the larger deconvolution problem.
The more general case is sometimes more akin to the case of the balls of separate radii r1
and r2, as found in Theorem 3.2. In this case the zero sets may coalesce, and furthermore
there may be issues related to the size of the functions at the zero sets. We investigate this case by forming the operator-valued transforms
π(ξ,η)(Tr1) = c Jn(r1pξ12+ · · · + ξ2n+ η12+ · · · + η2n) (r1pξ12+ · · · + ξn2 + η12+ · · · + η2n)n = Z r1 0 Jn−1(ρpξ12+ · · · + ξn2+ η21+ · · · + ηn2) (ρpξ2 1 + · · · + ξn2 + η12+ · · · + η2n)n−1 ρ2n−1dρ, and π(ξ,η)(Tr2) = c Jn(r2pξ12+ · · · + ξ2n+ η12+ · · · + η2n) (r2pξ12+ · · · + ξn2 + η12+ · · · + η2n)n = Z r2 0 Jn−1(ρpξ12+ · · · + ξn2+ η21+ · · · + ηn2) (ρpξ2 1 + · · · + ξn2 + η12+ · · · + η2n)n−1 ρ2n−1dρ.
Condition 1. of Theorem 3.2 is equivalent to the lack of a common kernel for these two representations. For the operator-valued functions jn(rip|λ|(P2+ Q2)), we still need to
expand to the representations π±λ(Tri) using the series (3) above. As in the above case of
the group Fourier transform of Tr, we compute
π±λ(Tri) = c ∞ X j=0 (−1)j Z ri 0 e−|λ|ρ24 L(n−1) j |λ|ρ2 2 ρ2n−1dρ e−(P2+Q2)L(n−1)j (2(P2+ Q2)) = c ∞ X j=0 (−1)j 2 n−1 |λ|nΨ (n−1) j |λ|r2 i 2 e−(P2+Q2)L(n−1)j (2(P2+ Q2)),
which similarly has the Hermite functions Eα as eigenfunctions. Thus the operator π±λ(Tri)
has the eigenfunction Ek in its kernel for any (λ, k) ∈ R∗ × Z+ such that
2n−1 |λ|nΨ (n−1) k (|λ|r 2 i/2) = Z ri 0 e−|λ|ρ2/4L(n−1)k (|λ|ρ2/2)ρ2n−1dρ = 0.
Thus we see condition 2. of Theorem 3.2 corresponds to no common kernel of π±λ(Tr1)
and π±λ(Tr2) for λ ∈ R
∗. Now ker{j
n(r1p|λ|(P2+ Q2))} ∩ ker{jn(r2p|λ|(P2+ Q2))} = {0}
implies that conditions 1. and 2. of Theorem 3.2 are met, which in turn implies the existence of the sequences of deconvolvers. This completes the proof of the theorem.
Note, however, that the lack of common zeros provided by the unified condition on the operator-valued Bessel functions of Theorem 4.1 address only part of the larger problem of deconvolution. This point can be seen from the analogous results for Euclidean spaces where the issue for deconvolution of the Pompeiu problem for two balls of appropriate radii divides into two separate cases, based on arithmetic conditions associated to the radii. As observed in [1], the separate conditions for the Bessel and Laguerre parts of the spectrum are unified by addressing the common kernel of the operator-valued transforms. However, at the level of the transform of the associated Euclidean space, the results of [6, 7, 8] demonstrate that for the problem of deconvolution and application of H¨ormander’s strongly coprime condition it is necessary to divide into two cases related to the separation of these zeros and how rapidly they can coalesce. The condition dividing these cases is how well the quotient of radii r1
r2
can be approximated by quotients of zeros p/q of the Bessel function Bn found in these
transforms. We require the following definition for N -well approximation. First, let En be
an infinite set with elements ordered by
En = {λ1, λ2, . . . , λn, . . .} where λj < λj+1.
Definition 1. For N > 0, a positive number α is called N -well approximated by ratios of En if, for every ` > 1, there exist indices j, k such that
|α − λk/λj| ≤ 1/(`jN).
If for every N > 0 the number α is not N -well approximated by ratios of En, then α is
called poorly approximated by ratios of En. We mention a result of Berenstein and Gay [5,
Proposition 6] demonstrating that when r1
r2 is not N -well approximated by ratios of En, then
b
µr2 satisfies the estimate
|µbr2(λk)| ≥ C/k
N +(n−1)/2
at λk, the zeros ofbµr1. From this result it is easy to see that if radii r1 and r2 that are poorly
approximated by zeros of Jn it can be shown that H¨ormander’s strongly coprime condition
holds, implying the existence of compactly supported deconvolvers. When the radii r1 and
r2 are N -well approximated H¨ormander’s strongly coprime condition is not met, implying
any deconvolvers cannot be compactly supported. Thus the distribution of these zero sets is integrally related to the issues of H¨ormander’s strongly coprime condition and the issue of deconvolution.
Note also that the procedure we have used in the above two results strongly suggests that the Bessel zeros are where the issue lies. When we can describe these zeros and find suitable neighborhoods to separate them, then it appears we can extend to a full neighborhood of the Bessel ray. What remains would then be only a finite number of Laguerre zeros, and a finite number of zeros should never introduce difficulties. Note this assumes no common
zeros, which assumption must be made in order to have injectivity. However, as we observe in the next section, this set of requirements refers to the type of deconvolution in the sense of limits as given in Section 3. Somewhat more will be required for the stronger form of deconvolution, in the sense of [6, 7], where a set of deconvolvers ν1, . . . , νnthat are compactly
supported distributions are shown to exist. We address these issues in the next section. 5. Convergence of Deconvolving Sequences
In this section we address the issue of applying the sequences of deconvolvers formed in Section 3 to perform the deconvolution. The problem of deconvolution for the Pompeiu problem can be interpreted to mean the reconstruction of a given function f from the integral information given in the Pompeiu problem, in this case representable using the convolutions f ∗T1 and f ∗T2. We observe in this section how the sequences {bν1,j} and {bν2,j} can be used to reconstruct the function f . We also address related issues including the appropriate spaces for the functions and distributions we are working with as well as issues of convergence. Finally we will relate these isssues to fundamental issues used in understanding deconvolution for Euclidean space, the Paley-Weiner theorem and H¨ormander’s strongly coprime condition.
We will utilize a theorem of Benson, Jenkins, and Ratcliff [2, Theorem 6.1] to describe the range of S(Hn) under the spherical function transform. In the notation of this theorem, the Heisenberg fan H is represented by 4(K, Hn), where K = U (n). The space bS(K, Hn)
consists of functions that are rapidly decreasing on 4(K, Hn). The space S
K corresponds
to radial functions in Schwartz space.
Theorem 4. ([2]) If f ∈ S(Hn) then bf ∈ bS(K, Hn). Conversely, if F ∈ bS(K, H
n), then
F = bf for some f ∈ S(Hn). Moreover, the map
b : S(Hb
n) → bS(K, Hn) is a bijection.
First notice that in the limit, our deconvolving sequences {eν1,j} and {νe2,j} allow us to construct the sequence of functions { efj} defined by
e
fj ≡ (f ∗ T1)e · eν1,j+ (f ∗ T2)e · eν2,j ≡ f · ee T1·
e
ν1,j + ef · eT2·eν2,j.
Noing that each efj has the property that efj|Kj = ef |Kj and efj|Fj = 0, as constructed above
in Section 3, we easily pass to the limit to attain the deconvolution e
f ≡ lim
j→∞(f ∗ T1)e · eν1,j+ (f ∗ T2)e · eν2,j.
This solves the problem of deconvolution, as f can be reconstructed from its “averages” f ∗ µ1 and f ∗ µ2. It is not as strong as the usual method of deconvolution, as we have here
used a limiting procedure. It now remains to discuss the convergence and to address the appropriate function spaces for both the deconvolving sequences and the associated sequences of functions. We further discuss the existence of limits ν1 = lim ν1,j and ν2 = lim ν2,j, forming
individual deconvolvers from the sequences.
Before passing to the limit, we consider the issue of the spherical function transform, showing the existence of sequences {ν1,j} and {ν2,j} whose spherical transforms yield the
deconvolving sequences we have constructed. The above theorem of [2] characterizing the image of S(Hn) under the spherical function transform will be used. At the level of the sequence of functions { efj}, as constructed above, we recognize that since efj|Kj ≡ 1 while
e
fj|Fj ≡ 0, efj is rapidly decreasing on 4(K, H
n) and thus ef
j ∈ bS(K, Hn). By the theorem
there exists Fj ∈ S(Hn) such that eFj ≡ efj. Furthermore note that these Fj in the sequence
can be explicitly constructed from the convolutions f ∗ T1 and f ∗ T2 using sequences of
Schwartz functions {ν1,j} and {ν2,j} as follows. We observe that νe1,j|Fj ≡ 0 and eν2,j|Fj ≡ 0
imply that eν1,j,eν2,j ∈ bS(K, Hn). This in turn implies the existence of the desired sequences of ν1,j, ν2,j ∈ S(Hn). Forming (f ∗ T1) ∗ ν1,j+ (f ∗ T2) ∗ ν2,j we see that
[(f ∗ T1) ∗ ν1,j+ (f ∗ T2) ∗ ν2,j]e = f · ee T1·eν1,j + ef · eT2·νe2,j = fej.
Thus (f ∗ T1) ∗ ν1,j + (f ∗ T2) ∗ ν2,j = Fj ∈ S(Hn). We will demonstrate below that the
sequence {Fj} approaches f in the limit.
Noting that Fj = f ∗ (T1∗ ν1,j+ T2∗ ν2,j), we now describe the appropriate space for the
elements eν1,j,eν2,j as well as T1∗ ν1,j, T2∗ ν2,j. Each of the (T1∗ ν1,j+ T2∗ ν2,j)e ∈ S(K, Hb n) since these were constructed to satisfy
(T1∗ ν1,j+ T2∗ ν2,j)e ≡ e T1·νe1,j+ eT2·νe2,j |Kj ≡ 1 |Fj ≡ 0,
and thus clearly are rapidly decreasing on 4(K, Hn). It follows from the theorem of [2] that
T1∗ ν1,j + T2∗ ν2,j ∈ S(Hn). A similar arguement shows each T1∗ ν1,j, T2∗ ν2,j ∈ S(Hn).
On the side of the spherical transform side we have constructed the eν1,j,eν2,j in order to deconvolve f ∗ T1 and f ∗ T2 on Kj. Letting φj = T1∗ ν1,j + T2∗ ν2,j, for each j we have
e φj|Kj = (T1∗ ν1,j+ T2∗ ν2,j)e|Kj = e T1·eν1,j+ eT2·eν2,j |Kj ≡ 1
so that in the limit
e
φj = (T1∗ ν1,j+ T2∗ ν2,j)e → 14(K,Hn).
We then recognize that 14(K,Hn) is a tempered distribution on 4(K, Hn). Furthermore, this
is the spherical function transform of the Dirac delta function δ ∈ S0(Hn). Since we know
that eδ(λ, k) = ψkλ(0) = 1 and eδ(0; ρ) = Jρ(0) = 1, we may write eδ = 14(K,Hn). Thus the
limit
(T1∗ ν1,j+ T2∗ ν2,j)e → 14(K,Hn) = eδ
tells us that limj→∞φj = limj→∞T1 ∗ ν1,j + T2 ∗ ν2,j = δ, by uniqueness of the Gelfand
transform. Also not that each φj ∈ S and that these converge to the tempered distribution
δ ∈ S0.
In the approach to deconvolution outlined in [8] the H¨ormander strongly coprime condition is used to demonstrate existence of deconvolvers ν1, . . . , νnas compactly supported
distribu-tions. The arithmetic condition for the radii of the disks not to be N -well approximated by the zeros of the Bessel function J1 is used to give the required estimates for the strongly
co-prime condition (2) near the Bessel zeros. The explicit construction of deconvolvers given in [6, 7] utilize a different set of conditions, different from but closely related to these. Our ap-proach is different from these in that we have produced sequences of deconvolvers {ν1,j} and
{ν2,j} in S(Hn). Since the transformations νe1,j and eν2,j are compactly supported for each j, we were able to use results on Schwartz space rather than H´ormander’s strongly coprime con-dition. In this section we have been considering how to utilize these deconvolving sequences to form the deconvolution through a limiting process. When we also address the issue of
the limits of the sequences of deconvolvers themselves, ν1 = lim ν1,j and ν2 = lim ν2,j, then
the Paley Weiner theorem and the strongly coprime condition of H¨ormander again become relevant. Considering the limit in the sense of distributions, we need to describe limhf, ν1,ji
for every f ∈ S(Hn). Due to the definition of ν
1,j in terms of its Gelfand transform eν1,j, we consider the limits h ef ,eν1,ji for ef ∈ bS, where the inner products h ef ,eν1,ji on the space Hn are to be interpreted using Godement’s Plancherel measure, as given in [2]. For convergence, it is necessary that νe1,j does not grow too rapidly. Due to the manner in which these were
constructed, this issue is directly related to the proximity of the zero sets of eT1 and eT2 and
their rates of decay near these zero sets. After setting up the limits and the issue of their convergence, we will briefly address three separate cases.
To discuss the limits of the deconvolving sequences {ν1,j} and {ν2,j}, we must consider the
limits in the sense of distributions. Since the deconvolving sequences have been defined in terms of the transforms eν1,j and νe2,j, we consider the limits of these sequences, as tempered distributions. Tempered distributions use the space of Schwartz functions as test functions, and this allows results to be transferred to ν1,j and ν2,j through the above result of [2]. Since
b
ν1,j ∈ bS ⊂ bS0, we have that
hf, ν1,ji = h ef ,eν1,ji for all ef ∈ bS
and we want to investigate the behavior in the limit as j → ∞. If we can show the limit converges for each bf ∈ bS, this will imply the existence of bν1 = limbν1,j and ν1 = lim ν1,j as tempered distributions, in bS0 and S0, respectively. However, this condition depends on the
rate of decay, or growth, of eν1,j. Recall that the definition ofνe1,j and νe2,j requires inversion of eT1 and eT2 away from their zeros. Although νe1,j,νe2,j ∈ bS for each j, depending on the proximity of the zeros of eT1 and eT2 and the growth ofνe1,j andνe2,j near these zeros, the limit of eν1,j andeν2,j may not remain in Schwartz space. Even if theeν1,j grow rapidly near the zero sets as j increases, the limit still exists in the space D0 since we know that for each f ∈ D, there exists k such that
limhf,eν1,ji = hf,eν1,ki,
where supp(f ) ⊂ Kk. Thus the rate of growth of the ν1,j and the space in which this
convergence occurs is directly related to the distribution of the zeros of eT1 and eT2, and these
issues also relate directly to the Paley-Weiner theorem. In Euclidean space Cn the strongly coprime condition of H¨ormander gives condition for existence ofνb1,bν2 as Fourier transforms of compactly supported distributions, or equivalently for the existence of ν1, ν2 as compactly
supported distributions. In the case of Section 3.1 the zeros have uniform separation, and it is easy to show H¨ormander’s strongly condition is satisfied. However in case Section 3.2, whether or not H¨ormander’s condition is satisfied is determined by whether the ration of radii r1
r2 is N -well approximated or is poorly approximated by ratios of Bessel zeros En,
where
En = {x ∈ R : Jn(x) = 0} = {λ1, λ2, . . . , λn, . . .}, where λj < λj+1.
In the first two cases, where the zeros have a uniform separation for the ball and sphere of Section 3.1 or the radii of the two balls of Section 3.2 where the radii are poorly approximated by ratios of En, it is possible to show the convergence exists in the space of
tempered distributins. However the third case in which the ratio or the radii of the two balls of Section 3.2 are N -well approximated, the convergence is more delicate and we can only guarantee limνe1,j =νe1 ∈ D
still valid in the space of tempered distributions, as discussed above. Note that the above conclusions were based on application of the results of [2] for Schwartz space and did not require stronger Paley-weiner type results. Ideally we would like to be able find methods to show existence of deconvolvers as compactly supported distribtions, or even to extend methods of [6, 7] to make an explicit construciton of such compactly supported deconvolvers using methods of summation, differentiation, integration, and convolution.
In the sequel to this paper we plan to revisit this issue and to deal more directly with the issue of H¨ormander’s strongly coprime condition in the Heisenberg group setting.
6. Extending Deconvolution from the Bessel Ray
Consider the case of radial distributions T1, . . . , Tncompactly supported satisfying H¨ormander’s
strongly coprime condition for Cn, implying the existence of distributions ν
1, . . . , νn radial
and compactly supported such that b
T1(ξ)νb1(ξ) + · · · + bTn(ξ)νbn(ξ) ≡ 1, which can be written as
b
T1(r)νb1(r) + · · · + bTn(r)νbn(r) ≡ 1,
where r = |ξ|. Noting that bTj(|ξ|) = bTj(r) = eTj(0; ρ) and likewise νbj(|ξ|) =νbj(r) =eνj(0; ρ), this is equivalent to a deconvolution of T1, . . . , Tn on the Bessel ray Hρ,
(6.1) Te1(0; ρ)
e
ν1(0; ρ) + · · · + eTn(0; ρ)eνn(0; ρ) ≡ 1.
It is important to ask whether such a deconvolution can be extended to the Gelfand trans-forms eT1, . . . , eTnon all of the Heisenberg brush H. It is not guaranteed that the deconvolvers
on Euclidean space will extend to work for the Heisenberg group. Considering the Gelfand transform of the same sum, eT1eν1+ · · · + eTneνn, the goal is to make this expression uniformly equal to 1 for all (λ, k) ∈ R∗× Z+,
(6.2) Te1(λ, k)eν1(λ, k) + · · · + eTn(λ, k)eνn(λ, k) ≡ 1.
The existence of ν1, . . . , νnsatisfying (6.1) and (6.2) would solve the problem of deconvolution
for Hn; however a solution of (6.1) has not been shown to extend to (6.2).
Our general goal for a set of radial distributions T1, . . . , Tn is to extend the Euclidean
deconvolution
b
T1(ξ)bν1(ξ) + · · · + bTn(ξ)bνn(ξ) ≡ 1
to all of the Heisenberg brush H by expanding upon the method developed in Section 3. In this section, we develop a method for extending the deconvolution from the central Bessel ray Hβ to all of the Heisenberg fan H for any Tr and Sr satisfying H¨ormander’s strongly
coprime condition, or equivalently, satisfying condition (6.1). That is to say, given ν1 and ν2
satisfying
b
Tr(ξ) ·bν1(ξ) + bSr(ξ) ·νb2(ξ) ≡ 1 for all ξ ∈ C
n
we want to find µ1 and µ2 satisfying
e
Tr(p) ·µe1(p) + eSr(p) ·µe2(p) ≡ 1 for all p ∈ H,
equivalent to conditions (6.1) and (6.2), and furthermore µej(0; ρ) = µbj(r), where r = |ξ|. Note that the constructions in Section 3 do not quite solve this problem, since the ν1 and
