Abstract
Let be the generalized Heisenberg group. In this paper, we study the inversion of the Radon transforms on . Several kinds of inversion Radon transform formulas are established. One is obtained from the Euclidean Fourier transform; the other is derived from the differential operator with respect to the center variable . Also by using sub-Laplacian and generalized sub-Laplacian we deduce an inversion formula of the Radon transform on .
1. Introduction
In the past decade the research of Radon transform on the Euclidean space has made considerable progress due to its wide applications to partial differential equations, X-ray technology, radio astronomy, and so on. The basic theory and some new developments can be found in [1] by Helgason and the references therein. The combination of Radon transform and wavelet transform has proved to be very useful both on pure mathematics and its applications. Therefore, it is very meaningful to give the inversion formula of the Radon transforms by using various ways. The first result in the area is due to Holschneider who considered the classical Radon transform on the two-dimensional plane (see [2]). Rubin in [3, 4] extended the results in [2] to the -dimensional Radon transform on and totally geodesic Radon transforms on the sphere and hyperbolic space. Heisenberg group is a vital Lie group with the underlying . Strichartz [5] discussed the Radon transform on the Heisenberg group. Nessibi and Trimèche [6] obtained an inversion formula of the Radon transform on the Laguerre hypergroup by using the generalized wavelet transform. Afterwards, He and Liu studied the analogous problems on the Heisenberg group and Siegel type Lie group (see [7, 8]), and Rubin [9] achieved some new progress of the Radon transform on . In [10] the authors gave the definition of generalized Heisenberg group denoted by and dealt with some problems related to geometric analysis. In this paper, we investigate the inversion formulas of the Radon transform on the generalized Heisenberg group. From the Euclidean Fourier transform and group Fourier transform, we deduce inversion formulas of the Radon transform on associated with differential operators and generalized sub-Laplacian.
Let be an -dimensional vector, where are positive real constants for . We can turn into a non-Abelian group by defining the group operation as This group is called the generalized Heisenberg group and is denoted by . It is obvious that the generalized Heisenberg group becomes ordinary Heisenberg group if all for . For any -dimensional vectors , , we define For , , the following equalities are valid:(i);(ii);(iii). Therefore, (1) can be rewritten as Identify with ; the symplectic form can be expressed by , where and . We can write (1) by or In next section, we will introduce some facts of Fourier analysis on , which is useful to get our result.
2. Fourier Analysis on
We first state the definition of the Hermite polynomials. Let , , and ; the Hermite polynomials are defined by The normalized Hermite functions are then defined by These functions form an orthonormal basis for . For any fixed , it is easy to calculate from (6) that then Therefore, we define which also form the orthonormal basis for for any positive constant .
Let , , , and for . The higher dimensional Hermite functions denoted by can be obtained by taking tensor products: Then the family is an orthonormal basis for (see [11]).
We define then for any fixed and is also the orthonormal basis for . If and for , then . And are eigenfunctions of the Hermite operator ; that is, , where is Laplacian on , and .
For , let be the Schrödinger representation of , which acts on by By a direct computation and the law of ; we then obtain which indicates that is unitary. In addition, we deduce that is irreducible (see [12]).
Suppose that , the Fourier transform of is an operator-valued function acting on by If we write , where , and define then (15) can be rewritten as where is the Weyl transform. By the same argument of the theory of Weyl transforms on the Heisenberg group we have From this identity we obtain the Plancherel formula where and is the Hilbert-Schmidt norm of operators. For , the Parseval formula is where denotes the adjoint of . For any , , where . Thus, can be written as . We define the left-invariant vector fields , and on . These are given by and (see [10]). The vectors fields generate the Lie algebra of the generalized Heisenberg group.
The generalized Heisenberg sub-Laplacian is explicitly given by Also, a direct computation shows that We know that (see [11]). In dimension one, in fact, we have In dimensions, this equation together with (11) shows that is an eigenfunction of the Hermite operators in each variable: Therefore, for each , we have Let for ; then where and .
Therefore, are eigenfunctions of operator with eigenvalues .
Now we consider the space spanned by . It is clear that the dimension of is (see [13]). Then we have Let denote the orthogonal projection operator from to . For . Let be the subspace of such that Then we have Set ; then we can write the above decomposition as More details can be found in [14].
3. Inversion of the Radon Transform on
The Radon transform for a function on the generalized Heisenberg group is defined by where , , , , and . Clearly, when , that is, for , the above formula is just the Radon transform on the Heisenberg group .
Next, we will obtain some inversion formulas for the inverse Radon transform by means of the Euclidean Fourier transform, differential operators, and sub-Laplacian. We first consider the way of the Euclidean Fourier transform. In fact, Strichartz [5] had obtained the inverse Radon transform on by using Euclidean Fourier transform. However, he did not show on which space the formula holds. In this section, we not only find a subspace of on which the Radon transform is a bijection, but also give the inversion Radon transform on , where is the Schwartz space on .
Let denote the Euclidean Fourier transform with respect to the central variable alone and let denote the full Euclidean Fourier transform; that is, Because we have We define By argument analogous to [9], we also find that if and only if , and is an isomorphism from onto . The spaces and are regarded as Semyanistyi-Lizorkin type spaces that have many applications (see [6, 15]). We define an operator which is given by It is easy to know that is a bijection on . The inversion of is given by Now (35) reads as Therefore, we have an inversion formula of the Radon transform as follows.
Theorem 1. Let . Then one has
We can give another inversion formula of by using operator . First of all, we give the Fourier transform of Radon transform for a function .
Lemma 2. Let ; then where is the orthonormal basis for .
Proof. Because
then we have
Let denote the ordinary Fourier transform of on ; then we have
On the other hand, by the recursion formula of Hermite polynomials (see [16]) we can get
so we have
This completes the proof.
Let ; then Furthermore, By (41), we have We can verify that and .
However, we know that may not belong to for a function . We naturally hope to find a space on which the Radon transform is a bijection. Suppose that is a subspace of such that the Radon transform is a bijection. That is, if , then for all . From Lemma 2 and the Plancherel formula (19), we have Define the subspace of by Obviously, if , then . Furthermore, for any , we can find that , such that . In fact, we take satisfying . Since we can see that . This is to say that the Radon transform is a bijection from onto itself.
From the above discussion, we have the following.
Theorem 3. Let ; then
Next, we will give another inversion formula associated with the generalized Heisenberg group sub-Laplacian. In fact, a direct calculation shows that and consequently is the scaled Hermite operator, where and are the left-invariant vector fields on .
Because and are eigenfunctions of operator with eigenvalues , we have Set ; then we have from [17] where .
Let ; then we can get Write ; by (41), we can deduce Consequently, by Plancherel formula we obtain
Theorem 4. Suppose that ; then we have where .
We conclude this section by giving the inverse Radon transform with generalized sub-Laplacian.
Let be positive constants for . We define generalized sub-Laplacian by where are the left-invariant vector fields on . Let ; by an analogous computation, we have If we take , then In this case the Fourier transform of under the action of generalized sub-Laplacian of is the same as that of the sub-Laplacian on (see [11]). Write ; by Lemma 2, Plancherel formula, and (65), we obtain the theorem below.
Theorem 5. Suppose that . Then we have where . Especially, when , we have
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The work for this paper is supported by the National Natural Science Foundation of China (no. 11271091).