2015
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On the convergence of the homotopy analysis method to solve the system of partial differential equations
2
2
One of the efficient and powerful schemes to solve linear and nonlinear equations is homotopy analysis method (HAM). In this work, we obtain the approximate solution of a system of partial differential equations (PDEs) by means of HAM. For this purpose, we develop the concept of HAM for a system of PDEs as a matrix form. Then, we prove the convergence theorem and apply the proposed method to find the approximate solution of some systems of PDEs. Also, we show the region of convergence by plotting the Hsurface.
1

87
100


A.
Fallahzadeh
Department of Mathematics, Islamic Azad University, Central Tehran Branch,
PO. Code 1416894351, Iran
Department of Mathematics, Islamic Azad University
Iran


M. A.
Fariborzi Araghi
Department of Mathematics, Islamic Azad University, Central Tehran Branch,
PO. Code 1416894351, Iran
Department of Mathematics, Islamic Azad University
Iran


V.
Fallahzadeh
Department of Mathematics, Islamic Azad University, Arac Branch, Iran
Department of Mathematics, Islamic Azad University
Iran
Homotopy Analysis Method
System of partial differential equations
Hsurface
[[1] S. Abbasbandy, Homotopy analysis method for the Kawahara equation, Nonlinear Analysis: Real World Applications 11 (2010) 307312. ##[2] S. Abbasbandy, Solitary wave solutions to the modied form of CamassaHolm equation by means of the homotopy analysis method, Chaos, Solitons and Fractals 39 (2009) 428435. ##[3] J. Biazar, M. Eslami, A new homotopy perturbation method for solving system of partial dierential equations, Computers and Mathematics with Applications 62 (2011) 225234. ##[4] J. Biazar, M. Eslami, H. Ghazvini, Homotopy perturbation method for system of partial differential equations, International Journal of Nonlinear Sciences and Numerical simulations 8 (3) (2007) 411416. ##[5] M.A. Fariborzi Araghi, A. Fallahzadeh, On the convergence of the Homotopy Analysis method for solving the Schrodinger Equation, Journal of Basic and Applied Scientic Research 2(6) (2012) 60766083. ##[6] M.A. Fariborzi Araghi, A. Fallahzadeh, Explicit series solution of Boussinesq equation by homotopy analysis method, Journal of American Science, 8(11) (2012). ##[7] T. Hayat, M. Khan, Homotopy solutions for a generalized secondgrade uid past a porous plate. Nonlinear Dyn 42 (2005) 395405. ##[8] S.J. Liao, Beyond pertubation: Introduction to the homotopy Analysis Method, Chapman and Hall/CRC Press, Boca Raton, (2003). ##[9] S.J. Liao, Notes on the homotopy analysis method: some denitions and theorems, Communication in Nonlinear Science and Numnerical Simulation, 14 (2009) 983997. ##[10] P. Roul, P. Meyer, Numerical solution of system of nonlinear integrodiggerential equation by Homotopy perturbation method, Applied Mathematical Modelling 35 (2011) 42344242. ##[11] A. Sami Bataineh, M.S.M. Noorani, I.Hashim, Approximation analytical solution of system of PDEs by homotopy analysis method, Computers and Mathematics with Applications 55 (2008) 29132923. ##[12] F. Wang, Y. An, Nonnegative doubly periodic solution for nonlinear teleghraph system, J.math.Anal.Appl. 338 (2008) 91100. ##[13] A.M. Wazwaz, The variational iteration method for solving linear and nonlinear system of PDEs, Comput, Math, Appl 54 (2007) 895902. ##[14] W. Wu, Ch. Liou, Out put regulation of twotimescale hyperbolic PDE systems, Journal of Process control 11 (2001) 637647. ##[15] W. Wu, S. Liao, Solving solitary waves with discontinuity by means of the homotopy analysis method. Chaos, Solitons & Fractals, 26 (2005) 177185. ##[16] E. Yusufoglu, An improvment to homotopy perturbation method for solving system of linear equations, Computers and Mathematic with Applications 58 (2009) 22312235.##]
Stochastic averaging for SDEs with Hopf Drift and polynomial diffusion coefficients
2
2
It is known that a stochastic differential equation (SDE) induces two probabilistic objects, namely a difusion process and a stochastic flow. While the diffusion process is determined by the innitesimal mean and variance given by the coefficients of the SDE, this is not the case for the stochastic flow induced by the SDE. In order to characterize the stochastic flow uniquely the innitesimal covariance given by the coefficients of the SDE is needed in addition. The SDEs we consider here are obtained by a weak perturbation of a rigid rotation by random elds which are white in time. In order to obtain information about the stochastic flow induced by this kind of multiscale SDEs we use averaging for the innitesimal covariance. The main result here is an explicit determination of the coefficients of the averaged SDE for the case that the diffusion coefficients of the initial SDE are polynomial. To do this we develop a complex version of Cholesky decomposition algorithm.
1

101
114


M.
Alvand
Department of Mathematical Sciences, Isfahan University of Technology,
Isfahan, Iran
Department of Mathematical Sciences, Isfahan
Iran
Stochastic differential equation
stochastic ow
stochastic averaging
Cholesky decomposition
system of complex bilinear equations
[[1] N. Abourashchi, A. Yu Veretennikov. On stochastic averaging and mixing, Theory Stoch. Process. 16, (1), (2010) 111129. ##[2] M. Alvand, Constructing an SDE from its twopoint generator, Stoch. Dyn. DOI: 10.1142/S0219493715500252 ##[3] L. Arnold, Random Dynamical Systems, SpringerVerlag, 1998. ##[4] P. H. Baxendale, Stochastic averaging and asymptotic behaviour of the stochastic Duffing  Van der Pol equation, Stochastic Process. Appl. 113, No. 2 (2004) 235272. ##[5] ———, Brownian motion in the diffeomorphisms group, Compositio Math. 53, No.1 (1984) 1950. ##[6] P. Bernard, Stochastic averaging, Nonlinear Stochastic Dynamics, (2002) 2942. ##[7] M. I. Freıdlin, The factorization of nonnegative definite matrices, Teor. Verojatnost. i Primenen. 13 (1968) 375378. ##[8] Z. L. Huang and W. Q. Zhu, Stochastic averaging of quasigeneralized Hamiltonian systems, Int. J. Nonlinear Mech. No.44 (2009) 7180. ##[9] H. Kunita, Stochastic Flows and Stochastic Differential Equations, Cambridge University Press, 1990. ##[10] J. A. Sanders, F. Verhulst and J. Murdack, Averaging Methods in Nonlinear Dynamical Systems, 2nd edition, Springer, 2007. ##[11] R. B. Sowers, Averaging of stochastic flows: Twist maps and escape from resonance, Stochastic Process. Appl. No. 119, (2009) 35493582. ##[12] S. Wiggins, An Introduction to Applied Nonlonear Dynamical Systems and Chaos, second edition, SpringerVerlag, 2009. ##[13] W.Q. Zhu, Z.L. Huang and Y. Suzuki, Stochastic averaging and Lyapunov exponent of quasi partially integrable Hamiltonian systems, Int. J. Nonlinear Mech. No.37 (2002) 419437.##]
Second order linear differential equations with generalized trapezoidal intuitionistic Fuzzy boundary value
2
2
In this paper the solution of a second order linear differential equations with intuitionistic fuzzy boundary value is described. It is discussed for two different cases: coefficient is positive crisp number and coefficient is negative crisp number. Here fuzzy numbers are taken as generalized trapezoidal intutionistic fuzzy numbers (GTrIFNs). Further a numerical example is illustrated.
1

115
129


S. P.
Mondal
Department of Mathematics, National Institute of Technology, Agartala,
Jirania799046, Tripura, India
Department of Mathematics, National Institute
Iran


T. K.
Roy
Department of Mathematics, Indian Institute of Engineering Science and Technology,
Shibpur, Howrah711103, West Bengal, India
Department of Mathematics, Indian Institute
India
fuzzy set
fuzzy differential equation
generalized trapezoidal intutionistic fuzzy number
[[1] L. A. Zadeh, Fuzzy sets, Information and Control, 8, (1965) 338353. ##[2] D.Dubois, H.Parade, Operation on Fuzzy Number, International Journal of Fuzzy system, 9, (1978) 613626. ##[3] K. T. Atanassov, Intuitionistic fuzzy sets, VII ITKRs Session, Soa, Bulgarian, 1983. ##[4] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, Vol. 20, (1986) 8796. ##[5] L. A. Zadeh, Toward a generalized theory of uncertainty (GTU) an outline, Information Sciences 172, (2005) 140. ##[6] S. L. Chang, L. A. Zadeh, On fuzzy mapping and control, IEEE Transaction on Systems Man Cybernetics 2, (1972) 30 34. ##[7] D. Dubois, H. Prade, Towards fuzzy dierential calculus: Part 3, Dierentiation, Fuzzy Sets and Systems 8, (1982) 225233. ##[8] M. L. Puri, D. A. Ralescu, Dierentials of fuzzy functions, Journal of Mathematical Analysis and Application##91, (1983) 552558. ##[9] R. Goetschel, W. Voxman, Elementary fuzzy calculus, Fuzzy Sets and Systems 18, (1986) 3143. ##[10] O. Kaleva, Fuzzy dierential equations, Fuzzy Sets and Systems 24, (1987) 301317. ##[11] B. Bede, A note on twopoint boundary value problems associated with nonlinear fuzzy differential equations, Fuzzy Sets. Syst.157, (2006) 986989. ##[12] B. Bede,S. G. Gal, Generalizations of the differentiability of fuzzynumbervalued functions with applications to fuzzy differential equations, Fuzzy Sets Syst.151, (2005) 581599. ##[13] Y. ChalcoCano, H. RomnFlores, On the new solution of fuzzy dierential equations, Chaos Solitons Fractals 38, (2008) 112119. ##[14] B. Bede,I. J. Rudas and A. L. Bencsik, First order linear fuzzy dierential equations under generalized differentiability, Inf. Sci. 177, (2007) 16481662. ##[15] Y. ChalcoCano, M.A.RojasMedar,H.RomnFlores, Sobre ecuaciones differencial esdifusas, Bol. Soc. Esp. Mat. Apl. 41, (2007) 9199. ##[16] Y. ChalcoCano, H. RomnFlores and M. A. RojasMedar, Fuzzy dierential equations with generalized derivative, in:Proceedings of the 27th North American Fuzzy Information Processing Society International Conference, IEEE, 2008. ##[17] L. Stefanini, B. Bede, Generalized Hukuhara differentiability of intervalvalued functions and interval differential equations, Nonlinear Anal. 71, (2009) 13111328. ##[18] A. Khastan,J. J. Nieto, A boundary value problem for secondorder fuzzy dierential equations, Nonlinear Anal. 72, (2010) 35833593. ##[19] P. Diamond, P. Kloeden, Metric Spaces of Fuzzy Sets, World Scientic, Singapore, 1994. ##[20] B. Bede, S. G. Gal, Generalizations of the dierentiability of fuzzynumbervalued functions with applications to fuzzy differential equations, Fuzzy Set Systems, 151 (2005) 581599. ##[21] L. Stefanini, A generalization of Hukuhara difference for interval and fuzzy arithmetic, in: D. Dubois, M.A. Lubiano, H. Prade, M. A. Gil, P. Grzegorzewski, O. Hryniewicz (Eds.), Soft Methods for Handling Variability and Imprecision, in: Series on Advances in Soft Computing, 48 (2008). ##[22] L. Stefanini, B. Bede, Generalized Hukuhara differentiability of intervalvalued functions and interval differential equations, Nonlinear Analysis, 71 (2009) 13111328. ##[23] A. Armand, Z. Gouyandeh, Solving twopoint fuzzy boundary value problem using the variational iteration method, Communications on Advanced Computational Science with Applications, Vol. 2013, (2013) 110. ##[24] N. Gasilov, S. E. Amrahov, A. G. Fatullayev, Solution of linear dierential equations with fuzzy boundary values, Fuzzy Sets and Systems 257, (2014) 169183. ##[25] N. Gasilov, S. E. Amrahov, A. G. Fatullayev, A. Khastan, A new approach to fuzzy initial value problem ,18(2), (2014) 217225. ##[26] B. Bede, L. Stefanini, Generalized dierentiability of fuzzyvalued functions, Fuzzy Sets and Systems, 230 (2013) 119141. ##[27] R. RodrguezLpez, On the existence of solutions to periodic boundary value problems for fuzzy linear differential equations, Fuzzy Sets and Systems, 219, (2013) 126. ##[28] S. Melliani, L. S. Chadli, Introduction to intuitionistic fuzzy partial differential Equations, Fifth Int. Conf. on IFSs, Sofia, 2223 Sept. 2001. ##[29] S. Abbasbandy, T. Allahviranloo, Numerical Solution of Fuzzy Differential Equations by RungeKutta and the Intuitionistic Treatment, Journal of Notes on Intuitionistic Fuzzy Sets, Vol. 8, No. 3, (2002) 4353. ##[30] S. Lata, A.Kumar, A new method to solve timedependent intuitionistic fuzzy differential equation and its application to analyze the intutionistic fuzzy reliability of industrial system, Concurrent Engineering: Research and Applications, (2012) 18. ##[31] S. P. Mondal and T. K. Roy, First order homogeneous ordinary differential equation with initial value as triangular intuitionistic fuzzy number, Journal of Uncertainty in Mathematics Science (2014) 117. ##[32] S. P. Mondal. and T. K. Roy, System of Differential Equation with Initial Value as Triangular Intuitionistic Fuzzy Number and its Application, Int. J. Appl. Comput. Math, (2010). ##[33] L. C. Barros, L. T. Gomes, P. A Tonelli, Fuzzy differential equations: An approach via fuzzification of the derivative operator, Fuzzy Sets and Systems, 230, (2013) 3952. ##[34] M.R.Seikh, P.K.Nayak and M.Pal, Generalized Triangular Fuzzy Numbers In Intuitionistic Fuzzy Environment, International Journal of Engineering Research and Development, Volume 5, Issue 1 (2012) 0813. ##[35] H.J.Zimmerman, Fuzzy set theory and its applications, Kluwer Academi Publishers, Dordrecht (1991).##]
New characterizations of fusion bases and Riesz fusion bases in Hilbert spaces
2
2
In this paper we investigate a new notion of bases in Hilbert spaces and similar to fusion frame theory we introduce fusion bases theory in Hilbert spaces. We also introduce a new denition of fusion dual sequence associated with a fusion basis and show that the operators of a fusion dual sequence are continuous projections. Next we dene the fusion biorthogonal sequence, Bessel fusion basis, Hilbert fusion basis and obtain some characterizations of them. we study orthonormal fusion systems and Riesz fusion bases for Hilbert spaces. we consider the stability of fusion bases under small perturbations. We also generalized a result of PaleyWiener [16] to the situation of fusion basis.
1

131
142


F.
Aboutorabi Goudarzi
Department of Mathematics, Faculty of Science, Central Tehran Branch, Islamic Azad University, Tehran, Iran
Department of Mathematics, Faculty of Science,
Iran


M. S.
Asgari
Department of Mathematics, Faculty of Science, Central Tehran Branch, Islamic Azad University, Tehran, Iran
Department of Mathematics, Faculty of Science,
Iran
Fusion Frame
Riesz fusion basis
Exact fusion frame
Orthonormal fusion basis
[[1] M. S. Asgari, G. Kavian, Expansion of Bessel and gBessel sequences to dual frames and dual gframes, J. Linear and Topological Algebra, Vol. 02, No. 01, 2013, 51 57 ##[2] M. S. Asgari, New characterizations of fusion frames (frames of subspaces), Proc. Indian Acad. Sci. (Math. Sci.) 119 No. 3 (2009), 114. ##[3] M. S. Asgari, On the stability of fusion frames (frames of subspaces), Acta Math. Sci. Ser. B, 31(4), (2011), 16331642. ##[4] M. S. Asgari, Operatorvalued bases on Hilbert spaces, J. Linear and Topological Algebra, Vol. 02, No. 04, (2013), 201218. ##[5] P. G. Casazza and G. Kutyniok, Frames of subspaces, in Wavelets, Frames and Operator Theory (College Park, MD, 2003), Contemp. Math. 345, Amer. Math. Soc. Providence, RI, 2004, 87113. ##[6] O. Christensen, An Introduction to frames and Riesz Bases, Birkhauser, Boston, 2003. ##[7] I. Daubechies, A. Grossmann and Y. Meyer, Painless nonorthogonal expansions, J. Math. Phys. 27(1986), 12711283. ##[8] R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72,(2), (1952), 341366. ##[9] M. Fornasier, Quasiorthogonal decompositions of structured frames, J. Math. Anal. Appl. 289 (2004), 180 199. ##[10] P. Gavruta, On the duality of fusion frames, J. Math. Anal. Appl. 333 (2007), 871879. ##[11] J.R. Holub, Preframe operators, Besselian frames and nearRiesz bases in Hilbert spaces, Proc. Amer. Math. Soc. 122 (1994) 779785. ##[12] V. Kaftal, D. R. Larson and Sh. Zhang, Operatorvalued frames, Trans. Amer. Math. Soc. 361 (2009), 6349 6385. ##[13] S. S. Karimizad, Gframes, gorthonormal bases and gRiesz bases, J. Linear and Topological Algebra, Vol. 02, No. 01, 2013, 2533. ##[14] W. Rudin, Functional Analysis, McGrawHill. Inc, New York, (1991). ##[15] W. Sun, Gframes and GRiesz bases, J. Math. Anal. Appl. (2006), 322, 437452. ##[16] R. Young, An Introduction to Nonharmonic Fourier Series, Academic Press, New York, 2001.##]
On the boundedness of almost multipliers on certain Banach algebras
2
2
Almost multiplier is rather a new concept in the theory of almost functions. In this paper we discussion the boundedness of almost multipliers on some special Banach algebras, namely stable algebras. We also define an adjoint and extension for almost multiplier.
1

143
152


E.
AnsariPiri
Department of Pure Mathematics, Faculty of Mathematical Science, University of Guilan, Rasht, Iran
Department of Pure Mathematics, Faculty of
Iran


M.
Shams Yousefi
Department of Pure Mathematics, Faculty of Mathematical Science, University of Guilan, Rasht, Iran
Department of Pure Mathematics, Faculty of
Iran


S.
Nouri
Department of Pure Mathematics, Faculty of Mathematical Science, University of Guilan, Rasht, Iran
Department of Pure Mathematics, Faculty of
Iran
Almost multipliers
almost additive maps
dual map
stable normed algebras
[[1] E. AnsariPiri, S. Nouri, Almost multipliers and some of their properties, Preprint. ##[2] E. AnsariPiri, S. Nouri, Stable normed algebra, Priprint. ##[3] F. Birtal, Isomorphism and isometric multipliers, Proc. Amer. Math. Soc. (1962), no. 13, 204210. ##[4] B. Host, F. Parreau, Sur un probleme de I. Glicksberg: les ideauxfermes de type ni deM(G), Ann. Inst. Fourier (Grenoble) 28(1978), no.3, 143164. ##[5] S. Helganson, Multipliers of Banach algebras, Ann. Math ,64 (1956), 240254. ##[6] R. Larsen, Theory of Multipliers, Springer, Berlin, 1971. ##[7] K. B. Laursen, M. Mbekhta, Closed range multipliers and generalized inverses, Stud. Math. 107 (1993)126135. ##[8] T. Miura, G. Hirasawa, S. Takahasi, Stability of multipliers on Banach algebras, A.M.S. 45 (2004), 23772381. ##[9] A. Ulger, Multipliers with closed range on commutative semsimple Banach algebras, Stud. Math. 153 (2002), no. 1, 5980. ##[10] J. Wang, Multipliers of commutative Banach algebras, Pacic J. Math, 11(1961), 11311149. ##[11] Y. Zaiem, Operateurs de convolution d'image femee et unitesapproches, Bull. Sci. Math. 99 (1975), 6574.##]
sTopological vector spaces
2
2
In this paper, we have dened and studied a generalized form of topological vector spaces called stopological vector spaces. stopological vector spaces are dened by using semiopen sets and semicontinuity in the sense of Levine. Along with other results, it is proved that every stopological vector space is generalized homogeneous space. Every open subspace of an stopological vector space is an stopological vector space. A homomorphism between stopological vector spaces is semicontinuous if it is scontinuous at the identity.
1

153
158


M.
Khan
Department of Mathematics, COMSATS Institute of Information
Technology, Park Road, Islamabad, Pakistan
Department of Mathematics, COMSATS Institute
Pakistan
moiz@comsats.edu.pk


S.
Azam
Punjab Education Department, Pakistan
Punjab Education Department, Pakistan
Pakistan


S.
Bosan
Punjab Education Department, Pakistan
Punjab Education Department, Pakistan
Pakistan
sTopological vector space
Semiopen set
semiclosed set
semicontinuous mapping
scontinuous mapping
left (right) translation
generalized homeomorphism
generalized homogeneous space
[[1] S. M. Alsulami and L. A. Khan, Weakly Almost Periodic Functions in Topologicl Vector Spaces, Afr. Diaspora J. Math.. (N.S.), 15(2)(2013), 7686. ##[2] G. Bosi, J.C. Candeal,; E. Indurain,; M. Zudaire, Existence of Homogenous Representations of interval Orders on a Cone in Topological Vector Space, Social Choice and welfare, Vol.24 (2005), 4561. ##[3] D. E. Cameron and G. Woods, sContinuous and sOpen Mappings, preprint. ##[4] Y. Q. Chen, Fixed Points for Convex Continuous mappings in Topological Vector Space, American Mathematical Society, Vol. 129 (2001), 21572162. ##[5] S. T. Clark, A Tangent Cone Analysis of Smooth Preferences on a Topological Vector Space, Economic Theory, Vol.23 (2004), 337352. ##[6] S. G. Crossley, S.K. Hildebrand, Semiclosed sets and semicontinuity in topological spaces, Texas J. Sci., Vol. 22 (1971), 123126. ##[7] S. G. Crossley, S.K. Hildebrand, Semiclosure, Texas J. Sci. 22 (1971), 99112. ##[8] S. G. Crossley, S.K. Hildebrand, Semitopological properties, Fund. Math. 74 (1972), 233254. ##[9] L. Drewnowski, Resolution of topological linear spaces and continuity of linear maps., Anal. Appl. 335 (2) (2007), 11771195. ##[10] A. Grothendieck. Topological vector spaces. New York: Gordon and Breach Science Publishers, (1973). ##[11] D. H. Hyers, Pseudonormed linear spaces and Abelian groups, Duke Mathematical Journal, Vol. 5 (1939), 628634. ##[12] J. L. Kelly, General topology, Van Nastrand (New York 1955). ##[13] Kolmogro, Zur Normierbarkeit eines topologischen linearen Raumes, Studia Mathematica, Vol. 5 (1934), 2933. ##[14] N. Levine, Semiopen sets and semicontinuity in topological spaces, Amer. Math. Monthly, Vol. 70 (1963), 3641. ##[15] J. V. Neuman, On complete topological spaces, Transactions of American Mathematical Society, Vol. 37 (1935), 12. ##[16] T. Noiri, On semi continuous mappings, Atti. Accad. Naz. Lin. El. Sci. Fis. mat. Natur. 8(54)(1973), 210214. ##[17] A. P. Robertson, W.J. Robertson, Topological vector spaces. Cambridge Tracts in Mathematics. 53. Cambridge University Press, (1964). ##[18] J. V. Wehausen, Transformations in Linear Topological Spaces, Duke Mathematical Journal, Vol. 4 (1938), 157169.##]
On dual shearlet frames
2
2
In This paper, we give a necessary condition for function in $L^2$ with its dual to generate a dual shearlet tight frame with respect to admissibility.
1

159
163


M.
Amin khah
Department of Application Mathematics, Kerman Graduate University of High Technology, PO. Code 76315115, Iran
Department of Application Mathematics, Kerman
Iran


A.
Askari Hemmat
Department of Mathematics, Shahid Bahonar University of Kerman, PO. Code 76175133, Iran
Department of Mathematics, Shahid Bahonar
Iran


R.
Raisi Tousi
Department of Mathematics, Ferdowsi University of Mashhad, PO. Code 115991775, Iran
Department of Mathematics, Ferdowsi University
Iran
Dual shearlet frame
Bessel sequence
admissible shearlet
[[1] C. K. Chui, X. Shi, On a LittlewoodPaley identity and characterization of wavelets, Math. Anal. Appl. 177 (1993) 608626. ##[2] I. Daubechies, B. Han, Pairs of dual wavelet frames from any two renable functions, Constr. Appr.,to appear. ##[3] B. Han, On dual wavelet tight frames, Appl. Comput. Harmon. Anal. 4 (1997) 380413. ##[4] G. Kutyniok, D. Labate, Shearlets: Multiscale Analysis for Multivariate Data, Birkhauser, Basel, 2012.##]