2014
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57
A recursive construction of a class of finite normalized tight frames
2
2
Finite normalized tight frames are interesting because they provide decompositions in applications and some physical interpretations. In this article, we give a recursive method for constructing them.
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1
7


A.
Abdollahi
Department of Mathematics, College of Sciences, Shiraz University, Shiraz, Islamic
Republic of Iran
Department of Mathematics, College of Sciences,
Iran


M.
Monfaredpour
Department of Mathematics, College of Sciences, Shiraz University, Shiraz, Islamic
Republic of Iran
Department of Mathematics, College of Sciences,
Iran
Frames
Tight frames
[[1] J. J. Benedetto and M. Fickus, Finite normalized tight frames, Adv. Comput. Math., 18(2003), 357385.##[2] O. Christensen, Frames and bases, An introductory course. Birkhauser, Boston, 2008. ¨##[3] N. Cotfas and J. P. Cazeau, Finite tight frames and some applications, J. Phys. A: Math. Theor. 43 (2010)##193001. doi:10.1088/17518113/43/19/193001.##[4] R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series , Trans. Amer. Math. Soc., 71,(1952)##[5] D. Feng, L. Wang and Y. Wang, Generation of finite tight frames by Householder transformations, Adv. Comput.##Math., 24(2006), 294309.##[6] J. Kovacevic and A. Chebira, ˇ Life beyond bases:The advent of frames (Parts I and II), IEEE Signal Proces. Mag.,##24(4),(2007), 86104, 115125.##[7] D. F. Li and W. C. Sun, Expansion of frames to tight frames, Acta Math. Sinica, English Series, 25, (2)(2009),##[8] V. N. Malozemov and A. B. Pevnyi, Equiangular tight frames, J. Math. Sci., 157(6)(2009), 325.##[9] A. Safapour and M. Shafiee, Constructing Finite Frames via Platonic Solids, Iran. J. Math. Sci. Info., 7(1)(2012),##[10] E. Soljanin, ˇ Tight frames in quantum information theory, DIMACS Workshop on Source Coding and Harmonic##Analysis, Rurgers, New Jersey, May 2002.##[11] T. Strohmer and R. Heat, Grassmannian frames withnapplications to codingamd commucations, Appl. Comp.##Harm. Anal. 14(3)(2003), 257275.##[12] J. C. Tremain, Algorithmic constructions of unitary matrices and tight frames, arXiv:1104.4539v1 [math.FA] 23##[13] G. Zimmermann, Normalized tight frames in finite dimensions, Inter. Seri. Numer. Math., 137(2001), 249  252.##]
Dilation of a family of gframes
2
2
In this paper, we first discuss about canonical dual of gframe ΛP = {ΛiP ∈ B(H, Hi) : i ∈ I}, where Λ = {Λi ∈ B(H, Hi) : i ∈ I} is a gframe for a Hilbert space H and P is the orthogonal projection from H onto a closed subspace M. Next, we prove that, if Λ = {Λi ∈ B(H, Hi) : i ∈ I} and Θ = {Θi ∈ B(K, Hi) : i ∈ I} be respective gframes for non zero Hilbert spaces H and K, and Λ and Θ are unitarily equivalent (similar), then Λ and Θ can not be weakly disjoint. On the other hand, we study dilation property for gframes and we show that two gframes for a Hilbert space have dilation property, if they are disjoint, or they are similar, or one of them is similar to a dual gframe of another one. We also prove that a family of gframes for a Hilbert space has dilation property, if all the members in that family have the same deficiency.
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9
18


M.
Abdollahpour
Department of Mathematics, Faculty of Mathematical Sciences, University of Mohaghegh
Ardabili, Ardabil, Islamic Republic of Iran
Department of Mathematics, Faculty of Mathematical
Iran
gRiesz basis
GFrame
disjointnes
[[1] M. R. Abdollahpour: Dilation of dual gframes to dual gRiesz bases, Banach J. Math. Anal. in press.##[2] M. R. Abdollahpour and F. Bagarello, On some properties of gframes and gcoherent states, Nuovo Cimento##Soc. Ital. Fis. B, 125(11) (2010), 1327–1342.##[3] M. R. Abdollahpour and A. Najati, Besselian gframes and near gRiesz bases, Appl. Anal. Discrete Math., 5##(2011), 259270.##[4] M. R. Abdollahpour and A. Najati, gFrames and HilbertSchmidt operators, Bull. Iran. Math. Soc., 37(4)##(2011), 141155.##[5] P. G. Casazza, D. Han and D. R. Larson, Frames for Banach spaces, Contemp. Math., 247(1999), 149–182.##[6] O. Christensen, An Introduction to Frame and Riesz Bases, Birkhauser 2002.##[7] D. Han and D. R. Larson, Frames, bases and group representations, Mem. Amer. Math. Soc., 147(697)(2000).##[8] A. Najati, M. H. Faroughi and A. Rahimi, Gframes and stability of gframes in Hilbert spaces, Methods Func. Anal. Topology, 4(2008), 271–286.##[9] W. Sun, Gframes and GRiesz bases, J. Math. Anal. Appl., 322 (2006), 437–452.##]
Legendre wavelets method for numerical solution of timefractional heat equation
2
2
In this paper, we develop an efficient Legendre wavelets collocation method for well known timefractional heat equation. In the proposed method, we apply operational matrix of fractional integration to obtain numerical solution of the inhomogeneous timefractional heat equation with lateral heat loss and Dirichlet boundary conditions. The power of this manageable method is confirmed. Moreover, the use of Legendre wavelets is found to be accurate, simple and fast.
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19
31


M. H.
Heydari
Department of Mathematics, Faculty of Mathematics, Yazd University, Yazd, Islamic
Republic of Iran
Department of Mathematics, Faculty of Mathematics,
Iran


F. M.
Maalek Ghaini
Department of Mathematics, Faculty of Mathematics, Yazd University, Yazd, Islamic
Republic of Iran
Department of Mathematics, Faculty of Mathematics,
Iran


M. R.
Hooshmandasl
Department of Mathematics, Faculty of Mathematics, Yazd University, Yazd, Islamic
Republic of Iran
Department of Mathematics, Faculty of Mathematics,
Iran
Legendre wavelets
Operational matrix of fractional integration
Timefractional heat equation
[[1] A. Carpinteri and F. Mainardi, Fractals and Fractional Calculus in Continuum Mechanics, Springer Verlag,##Wien, New York, 1997.##[2] K. S. Miller and B. Ross,An Introduction to the Fractional Calculus and Fractional Differential Equations,##Wiley, New York, 1993.##[3] K. B. Oldham and J. Spanier, The Fractional Calculus. Academic Press, New York, 1974.##[4] I. Podlubny, Fractional Differential Equations. Academic Press, San Diego, 1999.##[5] I. Podlubny. Fractionalorder systems and fractionalorder controllers. Report UEF0394, Slovak Academy of##Sciences, Institute of Experimental Physics, Kosice, Slovakia, November 1994, 18p.##[6] R. Gorenflo and F. Mainardi, Fractional calculus: integral and differential equations of fractional order, in: A.##carpinteri, f. mainardi (eds.), fractals and fractional calculus in continuum mechanics, Springer, New York,##(1997), 223–276.##[7] W. R. Schneider and W. Wyess, Fractional diffusion and wave equations, J. Math. Phys., 30(1989), 134–144.##[8] U. Lepik, Solving pdes with the aid of twodimensional haar wavelets, Comput. Math. Appl., 61(2011), 1873–##[9] U. Anderson and B. Engquist, A contribution to waveletbased subgrid modeling, Appl. Comput. Harmon.##Model, 7(1999), 151–164.##[10] C. Cattani, Haar wavelets based technique in evolution problems, Chaos,Proc. Estonian Acad. Sci. Phys. Math,##1(2004), 45–63.##[11] N. Coult, Explicit formulas for wavelethomogenized coefficients of elliptic operators, Appl. Comput. Harmon.##Anal, 21(2001), 360–375.##[12] X. Chen, J. Xiang, B. Li, and Z. He, A study of multiscale waveletbased elements for adaptive finite element##analysis, Adv. Eng. Softw, 41(2010),196–205.##[13] G. Hariharan, K. Kannan, and K.R. Sharma, Haar wavelet method for solving fishers equation, Appl. Math.##Comput, 211(2)(2009), 284–292.##[14] P. Mrazek and J. Weickert, From twodimensional nonlinear diffusion to coupled haar wavelet shrinkage, J. Vis.##Commun. Image. Represent, 18(2007),162–175.##[15] W. Fan and P. Qiao, A 2d continuous wavelet transform of mode shape data for damage detection of plate##structures, Internat. J. Solids Structures, 46(2003), 6473–6496.##[16] J. E. Kim, G.W. Yang, and Y.Y. Kim, Adaptive multiscale waveletgalerkin analysis for plane elasticity problems and its application to multiscale topology design optimation, internat. j. solids structures, Comput. Appl.##Math., 40(2003), 6473–6496.##[17] Y. Shen and W. Li, The natural integral equations of plane elasticity problems and its wavelet methods, Appl.##Math. Comput., 150 (2)(2004),417–438.##[18] Z. Chun and Z. Zheng, Threedimensional analysis of functionally graded plate based on the haar wavelet##method, Acta. Mech. Solida. Sin., 20(2)(2007), 95–102.##[19] H. F. Lam and C. T. Ng, A probabilistic method for the detection of obstructed cracks of beamtype structures##using spacial wavelet transform,Probab. Eng. Mech., 23(2008), 239–245.##[20] J. Majak, M. Pohlak, M. Eerme, and T. Lepikult, Weak formulation based haar wavelet method for solving##differential equations, Appl. Math. Comput., 211(2009), 488–494.##[21] L. M. S. Castro, A. J. M. Ferreira, S. Bertoluzza, R.C. Patra, and J.N. Reddy, A wavelet collocation method for##the static analysis of sandwich plates ussing a layerwise theory, Compos. Struct., 92(2010), 1786–1792.##[22] M. H. Heydari, M. R. Hooshmandasl, M. F. Maalek Ghaini and F. Fereidouni, Twodimensional Legendre wavelets for solving fractional Poisson equation with Dirichlet boundary conditions Engineering Anal. Boun.##Elem., 37(2013),13311338.##[23] L. Nanshan and E. B. Lin, Legendre wavelet method for numerical solutions of partial differential equations,##Numer. Methods Partial Dierential Equations, 26(1):81–94, 2009.##[24] M. U. Rehman and R. A. Khan, The legendre wavelet method for solving fractional differential equations,##Commun. Nonlinear Sci. Numer. Simul., 227(2)(2009), 234–244.##[25] A. Kilicman and Z.A. Al Zhour, Kronecker operational matrices for fractional calculus and some applications,##Appl. Math. Comput., 187(1)(2007), 250–65.##[26] M. H. Heydari, M. R. Hooshmandasl, F. M. Maalek Ghaini and F. Mohammadi, Wavelet Collocation Method for##Solving Multiorder Fractional Differential Equations, J. Appl. Math., vol. 2012, Article ID 542401, 19 pages,##2012. doi:10.1155/2012/542401.##[27] A. M. Wazwaz, Partial differential equations and solitary waves theory, Springer, Chicago, 2009.##]
Application of Shannon wavelet for solving boundary value problems of fractional differential equations I
2
2
Fractional calculus has been used to model physical and engineering processes that are found to be best described by fractional differential equations. Therefore, a reliable and efficient technique as a solution is regarded. This paper develops approximate solutions for boundary value problems of differential equations with noninteger order by using the Shannon wavelet bases. Wavelet bases have different resolution capability for approximating of different functions. Since for Shannontype wavelets, the scaling function and the mother wavelet are not necessarily absolutely integrable, the partial sums of the wavelet series behave differently and a more stringent condition, such as bounded variation, is needed for convergence of Shannon wavelet series. With nominate Shannon wavelet operational matrices of integration, the solutions are approximated in the form of convergent series with easily computable terms. Also, by applying collocation points the exact solutions of fractional differential equations can be achieved by wellknown series solutions. Illustrative examples are presented to demonstrate the applicability and validity of the wavelet base technique. To highlight the convergence, the numerical experiments are solved for different values of bounded series approximation.
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33
42


K.
Nouri
School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P. O. Box:
193955746, Tehran, Islamic Republic of Iran
School of Mathematics, Institute for Research
Iran


N.
Bahrami Siavashani
Department of Mathematics, Faculty of Mathematics, Statistics and Computer Sciences,
Semnan University, P. O. Box: 35195363, Semnan, Islamic Republic of Iran.
Department of Mathematics, Faculty of Mathematics,
Iran
Boundary value problems Fractional differential equations Shannon wavelet Operational matrix
[[1] J. J. Benedetto, P. J. S. G. Ferreira, Modern Sampling Theory, Springer Science and Business Media, New York,##[2] Y. M. Chen, Y. B. Wu, Wavelet method for a class of fractional convectiondiffusion equation with variable##coefficients, J. Comput. Sci., 1 (2010), 146149.##[3] K. Diethelm, The Analysis of Fractional Differential Equations, SpringerVerlag, Berlin, 2010.##[4] V. J. Ervin, J.P. Roop, Variational formulation for the stationary fractional advection dispersion equation, Numer. Methods Partial Dierential Equations, 22 (2006), 558576.##[5] V. D. Gejji, H. Jafari, Solving a multiorder fractional differential equation, Appl. Math. Comput., 189 (2007)##[6] I. Hashim, O. Abdulaziz, S. Momani, Homotopy analysis method for fractional IVPs, Commun. Nonlinear Sci.##Numer. Simul., 14 (2009), 674684.##[7] J. He, Nonlinear oscillation with fractional derivative and its applications, International Conference on Vibrating Engineering, Dalian, China, (1998), 288291.##[8] J. He, Some applications of nonlinear fractional differential equations and their approximations, Bull. Sci.##Technol., 15 (1999), 8690.##[9] S. Hosseinnia, A. Ranjbar, S. Momani, Using an enhanced homotopy perturbation method in fractional differential equations via deforming the linear part, Comput. Math. Appl., 56 (2008), 31383149.##[10] M. Inc, The approximate and exact solutions of the space and timefractional Burgers equations with initial##conditions by variational iteration method, J. Math. Anal. Appl., 345 (2008), 476484.##[11] A. Kilicman, Z.A.A. Al Zhour, Kronecker operational matrices for fractional calculus and some applications,##Appl. Math. Comput., 187 (2007), 250265.##[12] K. Moaddy, S. Momani, I. Hashim, The nonstandard finite difference scheme for linear fractional PDEs in fluid##mechanics, Comput. Math. Appl., 61 (2011), 12091216.##[13] S. Momani, Z. Odibat, Analytical approach to linear fractional partial differential equations arising in fluid##mechanics, Phys. Lett. A, 355 (2006), 271279.##[14] Y. Nawaz, Variational iteration method and homotopy perturbation method for fourthorder fractional integrodifferential equations, Comput. Math. Appl., 61 (2011), 23302341.##[15] Z. Odibat, S. Momani, The variational iteration method: an efficient scheme for handling fractional partial##differential equations in fluid mechanics, Comput. Math. Appl., 58 (2009), 21992208.##[16] I. Podlubny, Fractional Differential Equations, Academic Press, San Diego, 1999.##[17] S. S. Ray, K. S. Chaudhuri, R. K. Bera, Analytical approximate solution of nonlinear dynamic system containing##fractional derivative by modified decomposition method, Appl. Math. Comput., 182 (2006), 544552.##[18] M. U. Rehman, R. A. Khan, A numerical method for solving boundary value problems for fractional differential##equations, Applied Mathematical Modelling, 36 (2012), 894907.##[19] A. Saadatmandi, M. Dehghan, A new operational matrix for solving fractionalorder differential equations,##Comput. Math. Appl., 59 (2010), 13261336.##[20] H. Saeedi, M. Mohseni Moghadam, Numerical solution of nonlinear Volterra integrodifferential equations of##arbitrary order by CAS wavelets, Appl. Math. Comput., 16 (2011), 12161226.##[21] H. Saeedi, M. Mohseni Moghadam, N. Mollahasani, G. N. Chuev, A CAS wavelet method for solving nonlinear##Fredholm integrodifferential equations of fractional order, Commun. Nonlinear Sci. Numer. Simul., 16 (2011),##11541163.##[22] N. H. Sweilam, M. M. Khader, R. F. AlBar, Numerical studies for a multiorder fractional differential equation,##Phys. Lett. A, 371 (2007), 2633.##[23] Q. Wang, Numerical solutions for fractional KdVBurgers equation by Adomian decomposition method, Appl.##Math. Comput., 182 (2006), 10481055.##[24] M. Zurigat, S. Momani, A. Alawneh, Analytical approximate solutions of systems of fractional algebraicdifferential equations by homotopy analysis method, Comput. Math. Appl., 59 (2010), 12271235.##]
Linear preservers of twosided matrix majorization
2
2
For vectors X, Y ∈ Rn, it is said that X is left matrix majorized by Y if for some row stochastic matrix R; X = RY. The relation X ∼` Y, is defined as follows: X ∼` Y if and only if X is left matrix majorized by Y and Y is left matrix majorized by X. A linear operator T : Rp → Rn is said to be a linear preserver of a given relation ≺ if X ≺ Y on Rp implies that T X ≺ TY on Rn. The linear preservers of ≺` from Rp to Rn are characterized before. In this parer we characterize the linear preservers of ∼` from Rp to Rn, p ≥ 3. In fact we show that the linear preservers of ∼` from Rp to Rn are the same as the linear preservers of ≺` from Rp to Rn, but for p = 2, they are not the same.
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43
50


F.
Khalooei
Department of Mathmatics, Faculty of Mathmatics, Shahid Bahonar University of Kerman,
Kerman, Islamic Republic of Iran
Department of Mathmatics, Faculty of Mathmatics,
Iran
Linear preservers
Matrix majorization
Row stochastic matrix
[[1] T. Ando, Majorization, doubly stochastic matrices, and comparison of eigenvalues, Linear Algebra Appl., 118##(1989), 163248.##[2] A. Armandnejad, F. Akbarzadeh and Z. Mohamadi,Row and columnmajorization on Mn,m, Linear Algebra##Appl., 437(2012), 10251032.##[3] L. B. Beasley, S.G. Lee and Y.H. Lee, A characterization of strong preservers of matrix majorization, Linear##Algebra Appl. 367 (2003), 341346. [4] R. Bhatia, Matrix Analysis, SpringerVerlag, New York, 1997.##[5] R. A. Brualdi and G. Dahl,Majorization classes of integral matrices, Linear Algebra Appl., 436(2012), 802813.##[6] A. M. Hasani and M. Radjabalipour,Linear preserver of matrix majorization, International Journal of Pure and##Applied Mathematics, 32(4) (2006), 475482.##[7] A. M. Hasani and M. Radjabalipour,On linear preservers of (right) matrix majorization, Linear Algebra Appl.,##423(23)(2007), 255261.##[8] F. Khalooei, M. Radjabalipour and P. Torabian, Linear preservers of left matrix majorization, Electron. J. Linear##Algebra, 17(2008), 304315.##[9] F. Khalooei and A. Salemi,The structure of linear preservers of left matrix majorization on Rp, Electron. J.##Linear Algebra, 18(2009), 8897.##[10] F. Khalooei and A. Salemi,Linear preservers of majorization, Iranian Journal of Mathematical Siences and##Informatics, 6(2) (2011), 4350.##[11] C. K. Li and E. Poon,Linear operators preserving directional majorization, Linear Algebra Appl., 325 (2001),##[12] F. D. Martınez Perıa, P. G. Massey, and L. E. Silvestre, Weak MatrixMajorization, Linear Algebra Appl.,##403(2005), 343368.##]
Property (T) for C*dynamical systems
2
2
In this paper, we introduce a notion of property (T) for a C∗ dynamical system (A, G, α) consisting of a unital C∗algebra A, a locally compact group G, and an action α on A. As a result, we show that if A has strong property (T) and G has Kazhdan’s property (T), then the triple (A, G, α) has property (T).
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51
62


H.
Abbasi
Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Islamic
Republic of Iran.
Department of Mathematics, Azarbaijan Shahid
Iran
afrouzi@umz.ac.ir


Gh.
Haghighatdoost
Department of Mathematics, Azarbaijan Shahid Madani University, Tabriz, Islamic
Republic of Iran.
Department of Mathematics, Azarbaijan Shahid
Iran


I.
Sadeqi
Faculty of Sciences, Sahand University of Technology, Tabriz, Islamic Republic of Iran.
Faculty of Sciences, Sahand University of
Iran
Kazhdan’s property (T) Hilbert bimodule C*dynamical system
[[1] M. B. Bekka, Property (T) for C∗algebras, Bull. London Math. Soc. 38 (2006), 857  867.##[2] B. Bekka, P. de la Harpe and A. Valette, Kazhdan’s Property (T), New Mathematical Monographs 11, Cambridge##University Press, Cambridge 2008.##[3] A. Connes and V. Jones, Property (T) for von Neumann algebras, Bull. London Math. Soc. 17 (1985), 51  62.##[4] ChiWai Leung and ChiKeung Ng, Property (T) and strong Property (T) for unital C∗algebras, J. Func. Anal.##256 (2009), 3055  3070.##[5] K. R. Davidson, C∗algebras by example, Fields Inst. Monograph 6, Amer. Math. Soc., Providence 1996.##[6] P. Jolissaint, Property (T) for pairs of topological groups, Enseign. Math., 215 (2005), 31  45.##[7] D. Kazhdan, Connection of the dual space of a group with the structure of its closed subgroups, Funct. Anal.##Appl. (1967), 63  65.##[8] G. J. Murphy, C∗algebras and operator theory, Academic Press, San Diego 1990.##[9] M. Takesaki, Theory of operator algebras, Springer, Berlin 2003.##[10] D. Williams, Crossed products of C∗algebras, Math. Surveys Monogr., 134, Amer. Math. Soc., Providence##]
Twowavelet constants for square integrable representations of G/H
2
2
In this paper we introduce twowavelet constants for square integrable representations of homogeneous spaces. We establish the orthogonality relations for square integrable representations of homogeneous spaces which give rise to the existence of a unique self adjoint positive operator on the set of admissible wavelets. Finally, we show that this operator is a constant multiple of identity operator when G is a semidirect product group of a unimodular subgroup K and a closed subgroup H.
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63
73


R. A.
Kamyabi Gol
Department of Mathematics, Center of Excellency in Analysis on Algebraic
Structures(CEAAS), Ferdowsi University Of Mashhad, Mashhad, Islamic Republic of Iran.
Department of Mathematics, Center of Excellency
Iran


F.
Esmaeelzadeh
Department of Mathematics, Bojnourd Branch, Islamic Azad University, Bojnourd, Islamic
Republic of Iran.
Department of Mathematics, Bojnourd Branch,
Iran


R.
Raisi Tousi
Department of Mathematics, Ferdowsi University Of Mashhad, Mashhad, Islamic Republic
of Iran.
Department of Mathematics, Ferdowsi University
Iran
Homogenous space Irreducible representation
[[1] S.T. Ali, JP. Antoine, JP. Gazeau, Coherent States, Wavelets and Their Generalizations, SpringerVerlag, New##York, 2000.##[2] F. Esmaeelzadeh, R. A. Kamyabi Gol, R. Raisi Tousi, On the continuous wavelet transform on homogeneous##spaces, Int. J. Wavelets. Multiresolut, 10(4)(2012).##[3] F. Esmaeelzadeh, R. A. Kamyabi Gol, R. Raisi Tousi, Localization operators on Homogeneous spaces, Bull.##Iranian Math. Soc, Vol. 39, no. 3 (2013), 455467.##[4] F. Esmaeelzadeh, R. A. Kamyabi Gol, R. Raisi Tousi, Positive type functions and representations of some homogenous spaces , submitted.##[5] G.B. Folland, A Course in Abstract Harmonic Analysis, CRC Press, 1995.##[6] J.M.G. Fell, R.S. Doran, Representation of ∗algebras, Locally Compact Groups, and Banach ∗algebraic##Bundles, Vol. 1, Academic Press, 1988.##[7] H. Fuhr, ¨ Abstract Harmonic Analysis of Continuous Wavelet Transform, Springer Lecture Notes in Mathematics,##Nr. 1863, Berlin, 2005.##[8] A. Grossmann, J. Morlet, T. Paul, Transform associated to square integrable group representation I, J. Math.##Phys. 26 (1985), 2479.##[9] K.Grochenig, ¨ Foundations of TimeFrequency Analysis, Birkhauser Boston, 2001. ¨##[10] E. Hewitt, K.A. Ross, Abstract Harmonic Analysis, SpringerVerlag OHG. Gottingen. Heidelberg, 1963. ¨##[11] R.A. Kamyabi Gol, N. Tavallaei, Relatively invariant measure and semidirect product groups, submitted.##[12] R.A. Kamyabi Gol, N. Tavallaei, Wavelet transforms via generalized quasi regular representations, Appl. Comput. Harmon. Anal. 26 (2009), no. 3, 291300.##[13] R.A. Kamyabi Gol, N. Tavallaei, Convolution and homogeneous spaces, Bull. Iran. Math. Soc. 35(2009), 129##[14] T. Kato, Perturbation Theory for Linear Operator, SpringerVerlag, Berlin, 1976.##[15] M. Kyed, Square Integrable representations and the continuous wavelet transformation , Ph.D Thesis, 1999.##[16] H. Reiter, J. Stegeman, Classical Harmonic Analysis and Locally compact Group, clarendon press, 2000.##[17] W. Rudin, Functional Analysis, Mc GrawHill, 1974.##[18] M. W. Wong, Wavelet Transform and Localization Operators. Birkhauser Verlag, BaselBostonBerlin, 2002.##]