Research Article | Open Access
Jinsen Xiao, Jianxun He, "The Calderón Reproducing Formula Associated with the Heisenberg Group ", Mathematical Problems in Engineering, vol. 2010, Article ID 132186, 12 pages, 2010. https://doi.org/10.1155/2010/132186
The Calderón Reproducing Formula Associated with the Heisenberg Group
We establish the Calderón reproducing formula for functions in on the Heisenberg group . Also, we develop this formula in with .
The classical Calderón reproducing formula reads where , and denotes the convolution on The Calderón reproducing formula is a useful tool in pure and applied mathematics (see [1–4]), particularly in wavelet theory (see [5, 6]). We always call (1.1) an inverse formula of wavelet transform. In , the authors generalized (1.1) to when and are sufficiently nice normalized radial wavelet functions. The generalization of (1.1) involving nonradial wavelets and can be written in the following form: where and are rotated versions of and on . The authors in [8, 9] established (1.1) for Holschneider  studied the formula (1.2) in case and gave an inversion formula of the Radon transform in -space by using wavelets. Furthermore, Rubin  developed the Calderón reproducing formula, windowed X-ray transforms, the Radon transforms, and -plane transforms in -spaces on .
It is a remarkable fact that the Heisenberg group, denoted by , arises in two fundamental but different setting in analysis. On the one hand, it can be realized as the boundary of the unit ball in several complex variables. On the other hand, an important aspect of the study of the Heisenberg group is the background of physics, namely, the mathematical ideas connected with the fundamental nations of quantum mechanics. In other words, there is its genesis in the context of quantum mechanics which emphasizes its symplectic role in connection with the Fourier transform, pseudodifferential operators, and related matters (see ). Due to this reason, many interesting works were devoted to the theory of harmonic analysis on in [11–13] and the references therein. Also, the researches of wavelet analysis on are concerned increasingly; for this we refer readers to [14–16]. And the inversion formula of the Radon transform by using inverse wavelet transform on was established in . Our goal of the present article is to study the Calderón reproducing formula on the Heisenberg group in -space with . In the sequel we will develop the theory of inverse Radon transform on .
The Heisenberg group is a Lie group with the underlying manifold , and the multiplication law is given by where The dilation of is defined by with For , the homogeneous norm of is given by (see ) Notice that In addition, satisfies the quasitriangle inequality: The homogeneous dimension of is and the volume of a ball is , where is a constant.
Let ; then is a locally compact nonunimodular group with the group law The left and right Haar measures on are given by where denotes the Lebesgue measure on
Let be the space of measurable functions on , such that
Let and The Fock space is the space of holomorphic functions on such that From  we know that is an orthonormal basis of the Hilbert space For let be the Bargmann-Fock representation of which acts on by The group Fourier transform of a function is defined by
Let be the classes of Schatten-von Neumann operators on Hilbert space and let denote the algebra of all bounded operators, that is, For let denote the -norm of If is just the Hilbert-Schmit norm of that denotes Let denote the usual operator norm of in For let be the Banach space consisting of all weak measurable operator value functions , which also satisfy a.e. and For the Parseval formula is where denotes the adjoint of The Plancherel formula is As a consequence of (1.13), one has the inversion of the Fourier transform: Suppose and let By a direct computation, we have Let be the convolution of and that is, Then We should notice the following facts: if then And if , then The further detail of harmonic analysis on can be found in [11, 12].
2. Calderón Reproducing Formula
The authors in [14, 15, 18] studied the theory of continuous wavelet associated with the concept of square integrable group representations. The unitary representation of on is defined by Let denote the set of all positive real numbers, Let be the projection from to -dimensional subspace spanned by and let From [15, Theorem ], we have
Let and , ; if and satisfies then we call an admissible wavelet and write Let ; the continuous wavelet transform of with respect to is defined by And the following Calderón reproducing formula holds in the weak sense:
2.1. Calderón Reproducing Formula in
For let then We are now in a position to show that converges to in -space when and The result in this paper is an extension of that of Mourou and Trimèche .
Lemma 2.1. Suppose that and satisfies Let be defined by (2.10). Then one has
Theorem 2.2. Let and satisfy Then for one has and
2.2. Calderón Reproducing Formula in with
For with , the continuous wavelet transform of with respect to a wavelet can be defined by formula (2.7) under certain conditions on . In this part we will show that converges to
Let be a measurable function on ; for define It is easy to see that is nonnegative and radially decreasing, that is, and a.e. on
Let Then for any , we define and thus we have the following lemma.
Lemma 2.3. Let and be defined by (2.22). Then one has where is a positive constant, .
Proof. First we let It is obvious that for any Since is nonnegative and radially decreasing, from [11, page 542], we know that there exists a positive constant such that Thus, This completes the proof.
Let be a radial function in () and let for any , where is a unit vector of Then we have the following lemma.
Lemma 2.4. Let be a radial function in and let be defined by in (2.27). If satisfies and then
Proof. Analogous to (2.20) we define
and then a.e. on
Let and ; by (2.25) we have where is the Hardy-Littlewood maximal function of . From the definition of we have Because we get
By the hypothesis is radial and ; together with the definition of we have Since it follows from Lemma 2.3 that that is, On the other hand, and thus which implies that
The work for this paper is supported by the National Natural Science Foundation of China (no.s 10671041 and 10971039) and the Doctoral Program Foundation of the Ministry of Education of China (no. 200810780002).
- V. S. Guliyev and E. J. Ibrahimov, “Calderón reproducing formula associated with the Gegenbauer operator on the half-line,” Journal of Mathematical Analysis and Applications, vol. 335, no. 2, pp. 1079–1094, 2007.
- Y. Han, S. Lu, and D. Yang, “Inhomogeneous discrete Calderón reproducing formulas for spaces of homogeneous type,” Journal of Fourier Analysis and Applications, vol. 7, no. 6, pp. 571–600, 2001.
- M. Holschneider and P. Tchamitchian, “Pointwise analysis of Riemann's “nondifferentiable” function,” Inventiones Mathematicae, vol. 105, no. 1, pp. 157–175, 1991.
- B. Rubin, “The Calderón reproducing formula, windowed X-ray transforms, and Radon transforms in -spaces,” Journal of Fourier Analysis and Applications, vol. 4, no. 2, pp. 175–197, 1998.
- I. Daubechies, Ten Lectures on Wavelets, vol. 61 of CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, Pa, USA, 1992.
- Y. Meyer, Wavelets and Operators, vol. 37 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1992.
- M. Frazier, B. Jawerth, and G. Weiss, Littlewood-Paley Theory and the Study of Function Spaces, vol. 79 of CBMS Regional Conference Series in Mathematics, American Mathematical Society, Providence, RI, USA, 1991.
- M. A. Mourou and K. Trimèche, “Calderon's reproducing formula related to the Dunkl operator on the real line,” Monatshefte für Mathematik, vol. 136, no. 1, pp. 47–65, 2002.
- B. Rubin and E. Shamir, “Calderón's reproducing formula and singular integral operators on a real line,” Integral Equations and Operator Theory, vol. 21, no. 1, pp. 77–92, 1995.
- M. Holschneider, “Inverse Radon transforms through inverse wavelet transforms,” Inverse Problems, vol. 7, no. 6, pp. 853–861, 1991.
- E. M. Stein, Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, vol. 43 of Princeton Mathematical Series, Princeton University Press, Princeton, NJ, USA, 1993.
- D. Geller, “Fourier analysis on the Heisenberg group. I. Schwartz space,” Journal of Functional Analysis, vol. 36, no. 2, pp. 205–254, 1980.
- R. S. Strichartz, “ harmonic analysis and Radon transforms on the Heisenberg group,” Journal of Functional Analysis, vol. 96, no. 2, pp. 350–406, 1991.
- J. He and H. Liu, “Admissible wavelets associated with the affine automorphism group of the Siegel upper half-plane,” Journal of Mathematical Analysis and Applications, vol. 208, no. 1, pp. 58–70, 1997.
- 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. Liu, Y. Liu, and H. Wang, “Multiresolution analysis, self-similar tilings and Haar wavelets on the Heisenberg group,” Acta Mathematica Scientia B, vol. 29, no. 5, pp. 1251–1266, 2009.
- J. He, “An inversion formula of the Radon transform on the Heisenberg group,” Canadian Mathematical Bulletin, vol. 47, no. 3, pp. 389–397, 2004.
- J. He, “Continuous multiscale analysis on the Heisenberg group,” Bulletin of the Korean Mathematical Society, vol. 38, no. 3, pp. 517–526, 2001.
- M. A. Mourou and K. Trimèche, “Calderón's formula associated with a differential operator on and inversion of the generalized Abel transform,” Journal of Fourier Analysis and Applications, vol. 4, no. 2, pp. 229–245, 1998.
Copyright © 2010 Jinsen Xiao and Jianxun He. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.