2016
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Classical wavelet systems over finite fields
2
2
This article presents an analytic approach to study admissibility conditions related to classical full wavelet systems over finite fields using tools from computational harmonic analysis and theoretical linear algebra. It is shown that for a large class of nonzero window signals (wavelets), the generated classical full wavelet systems constitute a frame whose canonical dual are classical full wavelet frames as well, and hence each vector defined over a finite field can be represented as a finite coherent sum of classical wavelet coefficients as well.
1

1
18


Arash
Ghaani Farashahi
University of Vienna
University of Vienna
Iran
arash.ghaani.farashahi@univie.ac.at
Finite field
classical wavelet group
quasiregular representation
classical wavelet systems
classical dilation operators
[[1] A. Arefijamaal and E. Zekaee, Image processing by alternate dual Gabor frames, Bull. Iran. Math. Soc.,##42(6)(2016), 1305 1314.##[2] A. Arefijamaal and E. Zekaee, Signal processing by alternate dual Gabor frames, Appl. Comput. Harmon. Anal., 35(3)(2013), 535540.##[3] A. Arefijamaal and R.A. KamyabiGol, On the square integrability of quasi regular representation on semidirect product groups, J. Geom. Anal., 19(3)(2009), 541552.##[4] A. Arefijamaal and R.A. KamyabiGol, On construction of coherent states associated with semidirect products, Int. J. Wavelets Multiresolut. Inf. Process., 6(5) (2008), 749759.##[5] A. Arefijamaal and R.A. KamyabiGol, A Characterization of square integrable representations associated with CWT, J. Sci. Islam. Repub. Iran 18(2)(2007), 159166.##[6] I. Daubechies, The wavelet transform, timefrequency localization and signal analysis., IEEE Trans. Inform.##Theory, 36(5) (1990), 9611005.##[7] K. Flornes, A. Grossmann, M. Holschneider, and B. Torresani, Wavelets on discrete fields, Appl. Comput. Harmon. Anal., 1(2)(1994), 137146.##[8] A. Ghaani Farashahi, Structure of finite wavelet frames over prime fields, Bull. Iranian Math. Soc., to appear.##[9] A. Ghaani Farashahi, Wave packet transforms over finite cyclic groups, Linear Algebra Appl., 489(1) (2016), 7592.##[10] A. Ghaani Farashahi, Classical wavelet transforms over finite fields, J. Linear Topol. Algebra, 4 (4) (2015), 241257.##[11] A. Ghaani Farashahi, Wave packet transform over finite fields, Electron. J. Linear Algebra, 30 (2015), 507529.##[12] A. Ghaani Farashahi, Cyclic wavelet systems in prime dimensional linear vector spaces, Wavelets and Linear Algebra, 2 (1) (2015) 1124.##[13] A. Ghaani Farashahi, Cyclic wave packet transform on finite Abelian groups of prime order, Int. J. Wavelets##Multiresolut. Inf. Process., 12(6), 1450041 (14 pages), 2014.##[14] A. Ghaani Farashahi, M. MohammadPour, A unified theoretical harmonic analysis approach to the cyclic##wavelet transform (CWT) for periodic signals of prime dimensions, Sahand Commun. Math. Anal., 1(2)(2014),##[15] C. P. Johnston, On the pseudodilation representations of flornes, grossmann, holschneider, and torresani, J. Fourier Anal. Appl., 3(4)(1997), 377385.##[16] G. L. Mullen, D. Panario, Handbook of Finite Fields, Series, Discrete Mathematics and Its Applications, Chapman and Hall/CRC, 2013.##[17] R. J. McEliece, Finite Fields for Computer Scientists and Engineers, The Springer International Series in Engineering and Computer Science, 1987.##[18] G. Pfander, Gabor Frames in Finite Dimensions, In Finite Frames, Applied and Numerical Harmonic Analysis, 193239. Birkhauser Boston, 2013.##[19] O. Pretzel, ErrorCorrecting Codes and Finite Fields., Oxford Applied Mathematics and Computing Science##Series, 1996.##[20] R. Reiter and J.D. Stegeman, Classical Harmonic Analysis, 2nd Ed, Oxford University Press, New York, 2000.##[21] H. Riesel, Prime numbers and computer methods for factorization, (second edition), Boston, Birkhauser, 1994.##[22] S. A. Vanstone and P. C. Van Oorschot, An Introduction to Error Correcting Codes with Applications, The Springer International Series in Engineering and Computer Science, 1989.##[23] A. Vourdas, Harmonic analysis on a Galois field and its subfields, J. Fourier Anal. Appl., 14(1)(2008), 102123.##]
Linear combinations of wave packet frames for L^2(R^d)
2
2
In this paper we study necessary and sufficient conditions for some types of linear combinations of wave packet frames to be a frame for L2(Rd). Further, we illustrate our results with some examples and applications.
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19
32


Ashok
Sah
University of Delhi
University of Delhi
Iran
ashokmaths2010@gmail.com
Frames
Wave Packet Systems
Linear Combinations
[[1] A. Aldroubi, Portraits of frames, Proc. Amer. Math. Soc., 123(6)(1995), 1661–1668.##[2] P. G. Casazza, G. Kutyniok. Finite frames: Theory and Applications. Birkhauser, 2012.##[3] O. Christensen, Linear combinations of frames and frame packets, Z. Anal. Anwend., 20(4)(2001), 805–815.##[4] O. Christensen, An introduction to frames and Riesz bases, Birkhauser, Boston, 2002.##[5] O. Christensen, A. Rahimi, Frame properties of wave packet systems in L2(Rd), Adv. Compu. Math., 29(2008), 101–111.##[6] A. Cordoba, C. Fefferman, Wave packets and Fourier integral operators, Commun. Partial Differ. Equations,##3(11)(1978), 979–1005.##[7] W. Czaja, G. Kutyniok, D. Speegle, The geometry of sets of prameters of wave packets, Appl. Comput. Harmon. Anal., 20(1)(2006), 108–125.##[8] K. Guo, D. Labate, Some remarks on the unified characterization of reproducing systems, Collect. Math., 57 (3)(2006), 309–318.##[9] C. Heil, D. Walnut, Continuous and discrete wavelet transforms, SIAM Rev., 31(4)(1989), 628–666.##[10] C. Heil, A Basis Theory Primer, Expanded edition. Applied and Numerical Harmonic Analysis, Birkhauser,## Springer, New York, 2011.##[11] E. Hernandez, D. Labate, G. Weiss, A unified characterization of reproducing systems generated by a finite family II, J. Geom. Anal., 12(4)(2002), 615–662.##[12] E. Hernandez, D. Labate, G. Weiss, E. Wilson, Oversampling, quasiaffine frames and wave packets, Appl. Comput. Harmon. Anal., 16(2004), 111–147.##[13] D. Labate, G. Weiss, E. Wilson, An approach to the study of wave packet systems, Contemp. Math., 345(2004), 215–235.##[14] M. Lacey, C. Thiele, Lp estimates on the bilinear Hilbert transform for 2<p<1, Ann. Math., 146(1997),##693–724.##[15] M. Lacey, C. Thiele, On Calderons conjecture, Ann. Math., 149(1999), 475–496.##[16] A. K. Sah, L. K. Vashisht, Hilbert transform of irregular wave packet system for L2(R), Poincare J. Anal. Appl.,1(2014), 9–17.##[17] A. K. Sah, L. K. Vashisht, Irregular WeylHeisenberg wave packet frames in L2(R), Bull. Sci. Math., 139(1)(2015), 61–74.##]
Cartesian decomposition of matrices and some norm inequalities
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2
Let X be an nsquare complex matrix with the Cartesian decomposition X = A + i B, where A and B are n times n Hermitian matrices. It is known that $Vert X Vert_p^2 leq 2(Vert A Vert_p^2 + Vert B Vert_p^2)$, where $p geq 2$ and $Vert . Vert_p$ is the Schatten pnorm. In this paper, this inequality and some of its improvements are studied and investigated for the joint Cnumerical radius, joint spectral radius, and for the Cspectral norm of matrices.
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33
42


Alemeh
Sheikhhosseini
Department of Pure Mathematics, Shahid Bahonar University of Kerman, Kerman, Iran
Department of Pure Mathematics, Shahid Bahonar
Iran
sheikhhosseini@uk.ac.ir


Golamreza
Aghamollaei
Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman, Kerman, Iran
Department of Pure Mathematics, Faculty of
Iran
aghamollaei@uk.ac.ir
joint Cnumerical radius
Cspectral norm
joint spectral radius
[[1] Gh. Aghamollaei, N. Avizeh and Y. Jahanshahi, Generalized numerical ranges of matrix polynomials, Bull. Iran. Math. Soc., 39(5)(2013), 789–803.##[2] Gh. Aghamollaei, A. Salemi, Polynomial numerical hulls of matrix polynomials II, Linear Multilinear Algebra, 59(3)(2011), 291–302.##[3] R. Bhatia, Matrix Analysis, Springer, Berlin, 1997.##[4] R. Bhatia, T. Bhattacharyya, On the joint spectral radius of commuting matrices, Studia Math., 114(1)(1995), 29–37.##[5] R. Bhatia, F. Kittaneh, Cartesian decompositions and Schatten norms, Linear Algebra Appl., 318(13)(2012), 109–116.##[6] M.T. Chien, H. Nakazato, Joint numerical range and its generating hypersurface, Linear Algebra Appl., 432(1)(2010),173–179.##[7] R. Drnovsek, V. Muller, On joint numerical radius II, Linear Multilinear Algebra, 62(9)(2014): 1197–1204.##[8] R.A. Horn, C.R. Johnson, Matrix Analysis. Second Edition, Cambridge University Press, Cambridge, 2013.##[9] F. Kittaneh, M. S. Moslehian and T. Yamazaki, Cartesian decomposition and numerical radius inequalities,##Linear Algebra Appl., 471(4)(2015), 46–53.##[10] C.K. Li, Cnumerical ranges and Cnumerical radii, Linear Multilinear Algebra, 37(13)(1994), 51–82.##[11] C.K. Li and E. Poon, Maps preserving the joint numerical radius distance of operators, Linear Algebra Appl., 437(5)(2012), 1194–1204.##[12] C.K. Li, T. Tam and N. Tsing, The generalized spectral radius, numerical radius and spectral norm, Linear##Multilinear Algebra, 16(5)(1984), 215–237.##[13] A. Salemi and Gh. Aghamollaei, Polynomial numerical hulls of matrix polynomials, Linear Multilinear Algebra, 55(3)(2007), 219–228.##[14] B. Simon, Trace Ideals and their Applications, Cambridge University Press, Cambridge, New York, 1979.##]
Pseudoframe multiresolution structure on abelian locally compact groups
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2
Let $G$ be a locally compact abelian group. The concept of a generalized multiresolution structure (GMS) in $L^2(G)$ is discussed which is a generalization of GMS in $L^2(mathbb{R})$. Basically a GMS in $L^2(G)$ consists of an increasing sequence of closed subspaces of $L^2(G)$ and a pseudoframe of translation type at each level. Also, the construction of affine frames for $L^2(G)$ based on a GMS is presented.
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43
54


Hamide
Azarmi
Ph. D. student in Ferdowsi University of Mashhad
Ph. D. student in Ferdowsi University of
Iran
azarmi_1347@yahoo.com


Radjabali
Kamyabi Gol
Department of pure Mathematics; Ferdowsi University of Mashhad;
Department of pure Mathematics; Ferdowsi
Iran
kamyabi@um.ac.ir


Mohammad
Janfada
Department of pure Mathematics;Ferdowsi University of Mashhad;
Department of pure Mathematics;Ferdowsi University
Iran
janfada@um.ac.ir
Pseudoframe
generalized multiresolution structure
locally compact group, affine pseudoframe
[[1] J. J. Benedetto, S. Li, The theory of multiresolution analyses frames and applications to filter banks, Appl. Comp. Harm. Anal., 5 (1998), 389427.##[2] J. J. Benedetto, S. Li, Multiresolution analysis frames with applications, Proceeding ICASSP93 Proceedings of IEEE international conference on Acoustics, speech, and signal processing: digital speech processing Volume III Pages 304307, 1993.##[3] P. G. Casazza O. Christensen, D. Stoeva, Frame expansions in separable Banach space, J. Math. Anal. Appl., 114(1) (2005), 710–723.##[4] O. Christensen, On frame multiresolution analysis, In An Introduction to Frames and Riesz Bases, Part of the series Applied and Numerical Harmonic Analysis, 283311, 2003.##[5] O. Christensen, An Introduction to Frames and Riesz Bases, Birkhauser, Boston 2003.##[6] S. Dahlke, Multiresolution Analysis and Wavelets on Locally Compact Abelian Groups, Wavelets, Images, and Surface Fitting. P. J. Laurent, A. Le Mehaute, L. L. Schumaker, eds., A. K. Peters, Wellesley, 1994, 141156.##[7] I. Daubechies, Orthonormal bases of compactly supported wavelets, Commun. Pure Appl. Math., 41(9) (1988), 909996.##[8] I. Daubechies, A. Grossmann, Y. Meyer, Painless nonorthogonal expansion, J. Math. Phys. 27 (1986), 1271– 1283.##[9] R. Duffin R, S. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc., 72 (1952), 341–366.##[10] Yu. A. Farkov, Orthogonal wavelets on locally compact abelian groups, Funktsional. Anal. i Prilozhen., 31(4) (1997), 8688, English transl., Funct. Anal. Appl., 31 (1997), 294296.##[11] G. B. Folland, A Course in Abstract Harmonic Analysis, CRC Press, 1995.##[12] D. Gabor, Theory of communications, J. Inst. Electr. Eng., 93(26) (1946), 429–457.##[13] R. A. Kamyabi Gol, R. Raisi Tousi, The structure of shift invariant spaces on a locally compact abelian group, J. Math. Anal. Appl., 340(1) (2008), 219–225.##[14] R. A. Kamyabi Gol, R. Raisi Tousi, Some equivalent multiresolution conditions on locally compact abelian##groups, Proc. Math. Sci., 120(3) (2010), 317–331.##[15] S. V. Kozyrev, Wavelet theory as padic spectral analysis, Izv. Ross. Akad. Nauk Ser. Mat., 66(2) (2002), 149158, English transl, Izv. Math., 66 (2002), 367–376.##[16] W. C. Lang, Wavelet analysis on the Cantor dyadic group, Houston J. Math., 24(3)(1998), 533–544.##[17] S. Li, The theory of frame multiresolution analysis and its applications, Ph. D. Thesis, University of Maryland Graduate School, Baltimore, May 1993.##[18] S. Li, A theory of generalized multiresolution structure and pseudoframes of translates, J. Fourier Anal. and Appl. 7(1) (2001), 23–40.##[19] S. Li, H. Ogawa, A theory of peseudoframes for subspaces with applications, Proc. SPIE 3458, Wavelet Applications in Signal and Imaging Processing VI, 67(1998); doi:10.1117/12.328126.##[20] S. Li, H. Ogawa, Pseudoframes for subspaces with applications, J. Fourier Anal. Appl., 10(4)(2004), 409–431.##[21] S. G. Mallat, Multiresolution approximations and wavelet orthonormal bases of L2(R), Trans. Amer. Math. Soc., 315(1) (1989), 69–87.##[22] Y. Meyer, Wavelets and Operators, Translated by DH Salinger, Cambridge Studies in Advanced Mathematics, 1992.##[23] D. P. Petersen, D. Middleton, Sampling reconstruction of wavenumber limited functions in Ndimensional Euclidean spaces, Inf. Control, 5(4)(1962), 279–323.##]
Quartic and pantic Bspline operational matrix of fractional integration
2
2
In this work, we proposed an effective method based on cubic and pantic Bspline scaling functions to solve partial differential equations of fractional order. Our method is based on dual functions of Bspline scaling functions. We derived the operational matrix of fractional integration of cubic and pantic Bspline scaling functions and used them to transform the mentioned equations to a system of algebraic equations. Some examples are presented to show the applicability and effectivity of the technique.
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55
68


Ataollah
Askari Hemmat
Depatrment of Mathematics Graduate University of Advanced Technology
Depatrment of Mathematics Graduate University
Iran
askarihemmat@gmail.com


Tahereh
Ismaeelpour
Shahid Bahonar University of Kerman
Shahid Bahonar University of Kerman
Iran
tismaeelpour@math.uk.ac.ir


Habibollah
Saeedi
Shahid Bahonar University of Kerman, Kerman, Iran
Shahid Bahonar University of Kerman, Kerman,
Iran
saeedi@uk.ac.ir
Bspline
Wavelet
fractional equation
partial differential equation
Operational matrix of integration
[[1] M. Dehghan, J. Manafian, A. Saadatmandi, Solving nonlinear fractional partial differential equations using the homotopy analysis method, Numer. Methods Partial Differ. Equations, 26(2)(2010), 448479.##[2] J. Goswami, A. Chan, Fundamentals of wavelets theory, algorithms and applications, John Wiley and Sons, Inc., 1999.##[3] T. Ismaeelpour, A. Askari Hemmat and H. Saeedi, Bspline Operational Matrix of Fractional Integration, Optik International Journal for Light and Electron Optics, 130(2017), 291305.##[4] K. ALKhaled, Numerical solution of timefractional partial differential equations Using Sumudu decomposition method, Rom. J. Phys., 60(12)(2015),##[5] A. A. Kilbas, H. M. Srivastava and J.J. Trujillo, Theory and application of fractional differential equations.##NorthHolland Mathematics studies, Vol.204, Elsevier, 2006.##[6] M. Lakestani, M. Dehghan, S. Irandoustpakchin. The construction of operational matrix of fractional derivatives using Bspline functions, Commun. Nonlinear Sci. Numer. Simul., 17(3)(2012), 1149  1162.##[7] Y. Li, Solving a nonlinear fractional differential equations using chebyshev wavelets, Commun. Nonlinear. Sci. Numer. Simul., 15(9)(2010), 2284  2292.##[8] K.S. Miller, B. Ross, An introduction to the fractional calculus and fractional differential equations, Wiley,##New York, 1993.##[9] D. Sh. Mohammed, Numerical solution of fractional integrodifferential equations by least squares method and shifted Chebyshev polynomia, Math. Probl. Eng., 2014.##[10] I. Podlubny, Fractional differential equations. Academic Press, New York, 1999.##[11] H. Saeedi, Applicaion of Haar wavelets in solving nonlinear fractional Fredholm integrodifferential equations, J. Mahani Math. Res. Cent., 2 (1) (2013), 15  28.##[12] 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.##[13] M. Unser, Approximation power of biorthogonal wavelet expansions, IEEE Trans. Signal Process., 44 (39)##(1996), 519527.##[14] J.L. Wu, A wavelet operational method for solving fractional partial differential equations numerically, Appl. Math. Comput., 214 (1) (2009), 31  40.##]
Triangularization over finitedimensional division rings using the reduced trace
2
2
In this paper we study triangularization of collections of matrices whose entries come from a finitedimensional division ring. First, we give a generalization of Guralnick's theorem to the case of finitedimensional division rings and then we show that in this case the reduced trace function is a suitable alternative for trace function by presenting two triangularization results. The first one is a generalization of a result due to Kaplansky and in the second one a triangularizability condition which is dependent on a single element is presented.
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69
74


Hossein
Momenaee Kermani
Department of Pure Mathematics, Faculty of Mathematics and Computer, Shahid Bahonar University of Kerman
Department of Pure Mathematics, Faculty of
Iran
momenaee@uk.ac.ir
Triangularizable
Semigroup
Irreducible
Division ring
Reduced trace
[[1] D. Z. Docovic, B. H. Smith, Quaternionic matrices: Unitary similarity, simultaneous triangularization and some trace identities, Linear Algebra Appl., 428(4)(2008), 890910.##[2] P. K. Draxl, Skew fields, Cambridge University Press, 1983.##[3] R. M. Guralnick, Triangularization of sets of matrices, Linear Multilinear Algebra, 9(2)(1980), 133140.##[4] I. Kaplansky, The EngelKolchin theorem revisited. Contributions to Algebra, (Bass, Cassidy and Kovacik, Eds.), Academic Press, New York, 1977.##[5] T. Y. Lam, A first course in noncommutative rings. 2nd ed. Springer Verlag, New York, 2001.##[6] H. Momenaee Kermani, Triangularizability of algebras over division rings, Bull. Iran. Math. Soc., 34(1)(2008), 73  81.##[7] H. Momenaee Kermani, Triangularizability over ﬁelds and division rings. Ph. D. thesis, Shahid Bahonar University of Kerman, Kerman, Iran, 2005.##[8] M. Radjabalipour, P. Rosenthal and B. R. Yahaghi, Burnside's theorem for matrix rings over division rings,##Linear Algebra Appl., 382(2004), 2944.##[9] H. Radjavi, A trace condition equivalent to simultaneous triangularizability, Canada. J. Math., 38(1986), 376  386.##[10] H. Radjavi and P. Rosenthal, Simultaneous triangularization, SpringerVerlag, New York, 2000.##[11] W. S. Sizer, Similarity of sets of matrices over a skew field, Ph.D. thesis, Bedford college, University of London, 1975.##[12] B. R. Yahaghi, On Falgebras of algebraic matrices over a subfield F of the center of a division ring, Linear Algebra Appl., 418(23)(2006), 599613.##[13] B. R. Yahaghi, Reducibility results on operator semigroups. Ph.D. thesis, Dalhousie University, Halifax, N.S., Canada, 2002.##]