Document Type: Research Paper
Authors
- Mahdieh Fatemidokht ^{1}
- Ataollah Askari Hemmat ^{} ^{2}
^{1} Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran.
^{2} Department of Applied Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran.
Abstract
We introduced the continuous and discrete $p$-adic shearlet systems. We restrict ourselves to a brief description of the $p$-adic theory and shearlets in real case. Using the group $G_p$ consist of all $p$-adic numbers that all of its elements have a square root, we defined the continuous $p$-adic shearlet system associated with $L^2\left(Q_p^{2}\right)$. The discrete $p$-adic shearlet frames for $L^2\left(Q_p^{2}\right)$ is discussed. Also we prove that the frame operator $S$ associated with the group $G_p$ of all with the shearlet frame $SH\left( \psi; \Lambda\right)$ is a Fourier multiplier with a function in terms of $\widehat{\psi}$. For a measurable subset $H \subset Q_p^{2}$, we considered a subspace $L^2\left(H\right)^{\vee}$ of $L^2\left(Q_p^{2}\right)$. Finally we give a necessary condition for two functions in $L^2\left(Q_p^{2}\right)$ to generate a p-adic dual shearlet tight frame via admissibility.
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Main Subjects
[1] O. Christensen, An introduction to Frames and Riesz Bases, Birkhauser, Boston, 2003.
[2] S. Dahlke, G. Kutyniok, P. Maass, C. Sagiv, H.-G. Stark, and G. Teschke, The uncertainty principle associate with the continuous shearlet transform, Int. J. Wavelets Multiresolute. Inf. Process., 6 (2008), pp. 157-181.
[3] M. Fatemidokht and A. Askari Hemmat, $P$-adic shearlets, Wavel. Linear Algebra, to appear
[4] B. Han, On dual wavelet tight frames, Appl. Comput. Harmon. Anal., 4 (1997), pp. 380-413.
[5] G. Kutyniok and D. Labate, Construction of regular and irregular shearlet frames, J. Wavelet Theory Appl., 1 (2007), pp. 1-10.
[6] G. Kutyniok and D. Labate, Shearlets: Multiscle Analysis for Multivariate Data, Birkhauser. Basel, 2012.
[7] V.S. Valdimirov, I.V. Volovich, and E.I. Zelenov, $p$-Adic Analysis and Mathematical Physics, World Scientific, Singapore, 1994.