Research Article | Open Access
Multiresolution Analysis and Haar Wavelets on the Laguerre Hypergroup
Let be the Heisenberg group. The fundamental manifold of the radial function space for can be denoted by , which is just the Laguerre hypergroup. In this paper the multiresolution analysis on the Laguerre hypergroup is defined. Moreover the properties of Haar wavelet bases for are investigated.
In the past decade research on the multiresolution analysis has made considerable progress due to its wide applications. For the basic theory of multiresolution we refer readers to the work in [1, 2]. Recently, we find that a lot of authors try to extend the theory of wavelets on the Euclidean space to nilpotent Lie groups (see [3–6]).
In this paper we will give the definition of acceptable dilations on the Laguerre hypergroup. The multiresolution analysis on the Laguerre hypergroup is also defined. Moreover the properties of Haar wavelet bases for are investigated. We will prove the results analogous to those on in , on in , and on in .
Let be the positive measure defined on , for , by and denotes the space of all measurable functions on such that The generalized translation operator on is defined by for all , . It is said to be the Fourier transform of a function defined as follows: where , and the Laguerre function is defined on by , and is the Laguerre polynomial of degree and order . We know that for a pair of functions and , the generalized convolution product on the Laguerre hypergroup is defined by Further if and are in , then we have The functional analysis and Fourier analysis on and its dual have been extensively studied in [8, 9].
Let be a discrete subspace of . An automorphism is said to be an acceptable dilation for if it satisfies the following properties:(1) leaves invariant, that is, (2)all the eigenvalues, , of satisfy
The acceptable dilation on is defined by , for all Let be the dilation on the Laguerre hypergroup. Hence for all , . Clearly, for every and , is just an acceptable dilation on the Laguerre hypergroup. Now we give the definition of multiresolution analysis on the Laguerre hypergroup.
Definition 1.1 ((MRA(), , )). A multiresolution analysis on is an increasing sequence of closed subspaces of satisfying the following conditions:(1), (2)(3) for all (4)there exists a scaling function such that forms an orthonormal basis of .
From the above definition it is clear that is an orthonormal basis of . It follows from and that there exists a sequence such that The solution of (1.7) is often called a refinable function or a scaling function and is called a refinement sequence.
2. Acceptable Dilations on the Laguerre Hypergroup
In this section we will investigate the acceptable dilations on the Laguerre hypergroup. From the previous argument, we know that the acceptable dilations on the Laguerre hypergroup must satisfy three conditions:(1)they must be a automorphism of Laguerre hypergroup;(2)they must leave invariant;(3)the modulus of their eigenvalues must be more than 1.
Theorem 2.1. The acceptable dilations on must be the form where , , and , .
Proof. Let be the acceptable dilations on , where From the condition (1), we can obtain which implies that for all and . This yields and . From , we get By using the condition (3) we can obtain that and This concludes the proof of the theorem.
3. Multiresolution Analysis on the Laguerre Hypergroup
In this section, we only consider the dilation , where and . For simplicity we denote it by In order to obtain the main theorem, we need to give some lemmas to characterize the properties of the multiresolution analysis on .
Lemma 3.1. Suppose where and satisfies (2) and (4) of the definition of multiresolution analysis on the Laguerre hypergroup. The characteristic function of the set is a scaling function of multiresolution analysis. Then .
Proof. Let and . By using the property (4) of the Definition 1.1 we can obtain
which implies that . From (1.3), we know that and there exists a constant such that for all and . This yields , which implies that and cannot be nonzero at the same time.
Let . Then for any which implies that . Thus there exists a sequence such that This yields which implies that . Then we can see that Notice that If we let tend to infinity, then we can obtain . This implies that The desired result is thus obtained.
Lemma 3.2. Suppose , where and satisfies (2), (3), and (4) of the definition of multiresolution analysis on the Laguerre hypergroup. If the scaling function in (4) is in and , then .
Proof. Let and . Then we have
which implies . For , we can get the same result. It is easy to see that , for all and .
Let Then there exists a such that . For any , let and . Using , we immediately obtain . Then there exists a sequence such that , which implies Notice that , , and . Thus we can see that and , which implies for all and
Let . Then for any , there exists a such that . It follows from and that for all and .
For any , there must exist an element and such that is arbitrarily small, which implies that for any arbitrarily small . This yields for all and
Note and , This shows that Since , there exists some such that for all Let and . Then for all , which implies that for all , where . Then Notice that Thus we can see that which implies Let . Then and . This yields Taking into account the fact that when , we see when . Let tend to infinity, then for all which implies . Then We complete the proof of this theorem.
Theorem 3.3. Suppose is a scaling function for a multiresolution analysis associated with , where is the characteristic function of a measurable set . Then satisfies the following properties:(1), for a.e. , and ;(2)(3)(4) can be represented by the sequence where . Conversely, the characteristic function of a bounded measurable set that satisfies properties (1), (2), (3), and (4) is the scaling function of a multiresolution analysis associated with .
Proof. Suppose is a scaling function for a multiresolution analysis associated with . Then for all and which implies
Notice that and Thus we can obtain that , almost every By (1.7), we know that the second property is satisfied. Because of , we can see that . Let . Then . This implies . Therefore, can be represented by .
To see the converse, let Then is a family of closed subspace of . Let . Then Since , we can see that , which implies . This yields . Then we can also get . Notice that . Thus we can see that for all . Because can be represented by the sequence , thus .
In order to show that is a multiresolution analysis associated with , it suffices to show that and Further, it follows easily from Lemmas 3.1 and 3.2 that Our result is proved.
In this paper orthonormal Haar wavelet bases for are not constructed. But we believe that orthonormal Haar wavelet bases for can be constructed just as that in [2, 6, 7]. The details will appear elsewhere.
The second author is supported by the National Natural Science Foundation of China (no. 10671041) and the Doctoral Program Foundation of the Ministry of Education of China (no. 200810780002). The authors would be grateful to the referee for his/her invaluable suggestions.
- I. Daubechies, Ten Lectures on Wavelets, vol. 61 of CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 1992.
- K. Gröchenig and W. R. Madych, “Multiresolution analysis, Haar bases, and self-similar tilings of ,” IEEE Transactions on Information Theory, vol. 38, no. 2, part 2, pp. 556–568, 1992.
- D. Geller, “Fourier analysis on the Heisenberg group. I. Schwartz space,” Journal of Functional Analysis, vol. 36, no. 2, pp. 205–254, 1980.
- W. Lawton, “Infinite convolution products and refinable distributions on Lie groups,” Transactions of the American Mathematical Society, vol. 352, no. 6, pp. 2913–2936, 2000.
- H. Liu and L. Peng, “Admissible wavelets associated with the Heisenberg group,” Pacific Journal of Mathematics, vol. 180, no. 1, pp. 101–123, 1997.
- H. P. Liu, Y. Liu, and H. H. Wang, “Multiresolution analysis, self-similar tilings and Haar wavelets on the Heisenberg group,” to appear in Acta Mathematica Scientia Series B.
- P. Z. Xie and J. X. He, “Multiresolution analysis and Haar wavelets on the product of Heisenberg group,” International Journal of Wavelets, Multiresolution and Information Processing, vol. 7, no. 2, pp. 243–254, 2009.
- M. Assal and H. Ben Abdallah, “Generalized Besov type spaces on the Laguerre hypergroup,” Annales Mathématiques Blaise Pascal, vol. 12, no. 1, pp. 117–145, 2005.
- M. M. Nessibi and K. Trimèche, “Inversion of the Radon transform on the Laguerre hypergroup by using generalized wavelets,” Journal of Mathematical Analysis and Applications, vol. 208, no. 2, pp. 337–363, 1997.
Copyright © 2009 Peizhu Xie and Jianxun He. This is an open access article distributed under the