ORIGINAL_ARTICLE
On the characterization of subrepresentations of shearlet group
>We regard the shearlet group as a semidirect product group and show that its standard representation is,typically, a quasiregu- lar representation. As a result we can characterize irreducible as well as square-integrable subrepresentations of the shearlet group.
http://wala.vru.ac.ir/article_14265_9bf627ecedb9c35cc07168835acddb42.pdf
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1
9
Shearlet group
Semidirect product
V.
Atayi
true
1
Department of Pure Mathematics, Ferdowsi University of Mashhad,
Mashhad, Islamic Republic of Iran
Department of Pure Mathematics, Ferdowsi University of Mashhad,
Mashhad, Islamic Republic of Iran
Department of Pure Mathematics, Ferdowsi University of Mashhad,
Mashhad, Islamic Republic of Iran
LEAD_AUTHOR
R. A.
Kamyabi-Gol
true
2
Department of Pure Mathematics, Ferdowsi University of Mashhad,
Mashhad, Islamic Republic of Iran
Department of Pure Mathematics, Ferdowsi University of Mashhad,
Mashhad, Islamic Republic of Iran
Department of Pure Mathematics, Ferdowsi University of Mashhad,
Mashhad, Islamic Republic of Iran
AUTHOR
[1] G.S. Alberti, F. De Mari, E. De Vito and L. Mantovani, Reproducing Subgroups of S p(2, R). Part II: Admissible
1
Vectors, Monatsh. Math., 173(3)(2014), 261-307.
2
[2] S.T. Ali, J.P. Antoine, and J.P. Gazeau, Coherent States, Wavelets and Their generalizations. New York.
3
Springer-Verlag, 2000.
4
[3] J.P. Antoine, P. Carrette, R. Murenzi, and B. Piette, Image analysis with two-dimensional continuous wavelet
5
transform, Signal Process. 31(1993), 241-272.
6
[4] A.A. Arefijamaal, R.A. Kamyabi-Gol, A characterization of square integrable representat-ions associated with
7
CWT. J. Sci. Islam. Repub. Iran 18(2)(2007), 159-166.
8
[5] R.H. Bamberger and M.J.T. Smith, A filter bank for the directional decomposition of images: theory and design,
9
IEEE Trans. Signal Process., 40(1992) , 882-893.
10
[6] E.J. Candes and D.L. Donoho, ` New tight frames of curvelets and optimal representations of objects with piecewise C2 singularities, Comm. Pure and Appl. Math., 56 (2004), 216-266.
11
[7] S.H.H. Chowdhurya, S.T. Ali, All the groups of signal analysis from the (1+1)-affine Galilei group, J. Mathematical Physics, 52 (2011), 103-504.
12
[8] S. Dahlke, G. Kutyniok, P. Maass, C. Sagiv, H.-G. Stark, and G. Teschke, The uncertainty principle associated
13
with the continuous shearlet transform, Int. J. Wavelets Multiresolut. Inf. Process. 6 (2008), 157-181.
14
[9] S. Dahlke, G. Kutyniok, G. Steidl, and G. Teschke, Shearlet coorbit spaces and associated Banach frames.
15
Appl. Comput. Harm. Anal., 27(2) (2009), 195-214.
16
[10] S. Dahlke, G. Steidl, and G. Teschke, The continuous shearlet transform in arbitrary space dimensions. J.
17
Fourier Anal. Appl., 16(2010), 340-354.
18
[11] G.B. Folland, Real Analysis, John wiley, 1999.
19
[12] H. Fuhr, ¨ Abstract harmonic analysis of continuous wavelet transforms, vol. 1863 Lecture Notes in Mathematics.
20
Springer -Verlag, 2005.
21
[13] K. Guo, W.-Q Lim, D. Labate, G.Weiss and E. Wilson,Wavelets with composite dilations, Electron. Res. Announc. Amer. Math. Soc., 10(2004), 78-87.
22
[14] R.A. Kamyabi-Gol, V. Atayi, Abstract shearlet transform, preprint.
23
[15] N. Kingsbury, Complex wavelets for shift invariant analysis and filtering of signals, Appl. Computat. Harmon.
24
Anal., 10(2001), 234-253.
25
[16] N. Kingsbury, Image processing with complex wavelets, Phil. Trans. Royal Society London A, 357(1999), 2543-
26
[17] D. Labate, W.-Q. Lim, G. Kutyniok, and G. Weiss, Sparse multidimensional representation using shearlets,
27
Wavelets XI (San Diego, CA, 2005), 254-262, SPIE Proc. , 5914, Bellingham, WA., 2005.
28
[18] E.P. Simoncelli, W.T. Freeman, E.H. Adelson, D.J. Heeger, Shiftable multiscale transforms, IEEE Trans. Inform.
29
Theory, 38 (1992), 587-607.
30
ORIGINAL_ARTICLE
Cyclic wavelet systems in prime dimensional linear vector spaces
>Finite affine groups are given by groups of translations and di- lations on ﬁnite cyclic groups. For cyclic groups of prime order we develop a time-scale (wavelet) analysis and show that for a large class of non-zero window signals/vectors, the generated full cyclic wavelet system constitutes a frame whose canonical dual is a cyclic wavelet frame.
http://wala.vru.ac.ir/article_14266_785b6fb99f7fe9032e4a04c9dc31a079.pdf
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11
24
Cyclic wavelet system
Cyclic wavelet frame
A.
Ghaani Farashahi
true
1
Numerical Harmonic Analysis Group (NuHAG), Faculty of Mathematics,
University of Vienna
Numerical Harmonic Analysis Group (NuHAG), Faculty of Mathematics,
University of Vienna
Numerical Harmonic Analysis Group (NuHAG), Faculty of Mathematics,
University of Vienna
LEAD_AUTHOR
[1] A. Arefijamaal, R. Kamyabi-Gol, On the square integrability of quasi regular representation on semidirect
1
product groups, J. Geom. Anal. 19 (2009), no. 3, 541-552.
2
[2] A. Arefijamaal, R. Kamyabi-Gol, On construction of coherent states associated with semidirect products, Int. J.
3
Wavelets Multiresolut. Inf. Process. 6 (2008), no. 5, 749-759.
4
[3] A. Arefijamaal, R. Kamyabi-Gol, A characterization of square integrable representations associated with CWT,
5
J. Sci. Islam. Repub. Iran 18 (2007), no. 2, 159-166.
6
[4] G. Caire, R. L. Grossman and H. Vincent Poor, Wavelet transforms associated with finite cyclic Groups, IEEE
7
Trans. Information Theory, Vol. 39, No. 4, 1993.
8
[5] P. Casazza and G. Kutyniok. Finite Frames Theory and Applications. Applied and Numerical Harmonic Analysis. Boston, MA: Birkhauser. 2013.
9
[6] L. Cohen, Time-Frequency Analysis, Prentice-Hall, New York, 1995.
10
[7] I. Daubechies, Ten Lectures on Wavelets, Society for Industrial and Applied Mathematics, 1992.
11
[8] I. Daubechies and B. Han. The canonical dual frame of a wavelet frame. Appl. Comput. Harmon. Anal.,
12
12(3):269-285, 2002.
13
[9] R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Sci. 72 (1952)
14
[10] P. Flandrin, Time-Frequency/Time-Scale Analysis, Wavelet Analysis and its Applications, Vol. 10 Academic
15
Press, San Diego, 1999.
16
[11] K. Flornes, A. Grossmann, M. Holschneider and B. Torresani, ´ Wavelets on discrete fields, Appl. Comput. Harmon. Anal., 1(1994),137-146.
17
[12] G. B. Folland, A Course in Abstract Harmonic Analysis, CRC Press, 1995.
18
[13] S. Foucart and H. Rauhut, A mathematical introduction to compressive sensing, Applied and Numerical Harmonic Analysis. Springer, 2013.
19
[14] A. Ghaani Farashahi, Wave packet transforms over finite fields, to appear, 2015.
20
[15] A. Ghaani Farashahi, Cyclic wave packet transform on finite Abelian groups of prime order, Int. J. Wavelets
21
Multiresolut. Inf. Process., 12(6) (2014), 1450041, 14 pp.
22
[16] A. Ghaani Farashahi, Continuous partial Gabor transform for semi-direct product of locally compact groups,
23
Bull. Malays. Math. Sci. Soc., 38(2)(2015), 779-803.
24
[17] A. Ghaani Farashahi, R. Kamyabi-Gol, Gabor transform for a class of non-abelian groups, Bull. Belg. Math.
25
Soc. Simon Stevin 19 (2012), no. 4, 683-701.
26
[18] A. Ghaani Farashahi, M. Mohammad-Pour, A unified theoretical harmonic analysis approach to the cyclic
27
wavelet transform (CWT) for periodic signals of prime dimensions, Sahand Commun. Math. Anal., 1(2)(2014),1-
28
[19] G.H. Hardy and E.M. Wright, An Introduction to the Theory of Numbers, Oxford Univ. Press, 1979.
29
[20] B. Han, On dual wavelet tight frames, Appl. Comput. Harmon. Anal., 4 (1997), 380-413.
30
[21] E. Hewitt and K. A. Ross, Abstract Harmonic Analysis, Vol 1, 1963.
31
[22] C. P. Johnston, On the pseudodilation representations of flornes, grossmann, holschneider, and torr´esani, J.
32
Fourier Anal. Appl., 3(4)(1997), 377-385.
33
[23] G. L. Mullen and D. Panario, Handbook of Finite Fields, Series: Discrete Mathematics and Its Applications,
34
Chapman and Hall/CRC, 2013.
35
[24] G. Pfander. Gabor Frames in Finite Dimensions, In G. E. Pfander, P. G. Casazza, and G. Kutyniok, editors,
36
Finite Frames, Applied and Numerical Harmonic Analysis, 193-239. Birkhauser Boston, 2013.
37
[25] G. Pfander and H. Rauhut, Sparsity in time-frequency representations, J. Fourier Anal. Appl., 11(6)(2010), 715-
38
[26] G. Pfander, H. Rauhut, and J. Tropp, The restricted isometry property for time-frequency structured random
39
matrices, Probability Theory and Related Fields, 3-4(156)(2013), 707-737.
40
[27] G. Pfander, H. Rauhut, and J. Tanner, Identification of matrices having a sparse representation, IEEE Transactions on Signal Processing, 56 (11)(2008), 5376-5388.
41
[28] H. Rauhut, Compressive sensing and structured random matrices, In M. Fornasier, editor, Theoretical foundations and numerical methods for sparse recovery, volume 9 of Radon Series Comp. Appl. Math., pp 1-92.
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deGruyter, 2010.
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[29] H. Riesel, Prime Numbers and Computer Methods for Factorization (second edition), Boston: Birkhauser, ISBN
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0-8176-3743-5, 1994.
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[30] S. Sarkar, H. Vincent Poor, Cyclic Wavelet Transforms for Arbitrary Finite Data Lengths, Signal Processing, 80
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(2000), 2541-2552.
47
[31] G. Strang and T. Nguyen, Wavelets and Filter Banks, Wellesley-Cambridge Press, Wellesley, MA, 1996.
48
[32] M. Vetterli and J. Kovacevic, Wavelets and Subband Coding, Prentice Hall, ISBN 0-13-097080-8, 1995.
49
ORIGINAL_ARTICLE
On the distance from a matrix polynomial to matrix polynomials with two prescribed eigenvalues
>Consider an n × n matrix polynomial P(λ). A spectral norm distance from P(λ) to the set of n × n matrix polynomials that have a given scalar µ ∈ C as a multiple eigenvalue was introduced and obtained by Papathanasiou and Psarrakos. They computed lower and upper bounds for this distance, constructing an associated perturbation of P(λ). In this paper, we extend this result to the case of two given distinct complex numbers µ1 and µ2. First, we compute a lower bound for the spectral norm distance from P(λ) to the set of matrix polynomials that have µ1, µ2 as two eigenvalues. Then we construct an associated perturbation of P(λ) such that the perturbed matrix polynomial has two given scalars µ1 and µ2 in its spectrum. Finally, we derive an upper bound for the distance by the constructed perturbation of P(λ). Numerical examples are provided to illustrate the validity of the method.
http://wala.vru.ac.ir/article_14267_031c86cfbf3947aad1230b028a5506b5.pdf
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25
38
Matrix polynomial
Eigenvalue
Perturbation
Singular value
E.
Kokabifar
true
1
Faculty of Science, Yazd University, Yazd, Islamic Republic of Iran.
Faculty of Science, Yazd University, Yazd, Islamic Republic of Iran.
Faculty of Science, Yazd University, Yazd, Islamic Republic of Iran.
LEAD_AUTHOR
G.B.
Loghmani
true
2
Faculty of Science, Yazd University, Yazd, Islamic Republic of Iran.
Faculty of Science, Yazd University, Yazd, Islamic Republic of Iran.
Faculty of Science, Yazd University, Yazd, Islamic Republic of Iran.
AUTHOR
A. M.
Nazari
true
3
Department of Mathematics, Faculty of Science, Arak University, Arak,
Islamic Republic of Iran.
Department of Mathematics, Faculty of Science, Arak University, Arak,
Islamic Republic of Iran.
Department of Mathematics, Faculty of Science, Arak University, Arak,
Islamic Republic of Iran.
AUTHOR
S. M.
Karbassi
true
4
Department of Mathematics, Yazd Branch, Islamic Azad University, Yazd,
Islamic Republic of Iran.
Department of Mathematics, Yazd Branch, Islamic Azad University, Yazd,
Islamic Republic of Iran.
Department of Mathematics, Yazd Branch, Islamic Azad University, Yazd,
Islamic Republic of Iran.
AUTHOR
[1] J. W. Demmel, Applied Numerical Linear Algebra, SIAM, Philadelphia, 1997.
1
[2] J.W. Demmel, On condition numbers and the distance to the nearest ill-posed problem, Numer.Math., 51 (1987),
2
251–289.
3
[3] I. Gohberg, P. Lancaster and L. Rodman, Matrix Polynomials, Academic Press, New York, 1982.
4
[4] T. Kaczorek, Polynomial and Rational Matrices: Applications in Dynamical Systems Theory, Springer-Verlag,
5
London, 2007.
6
[5] P. Lancaster, Lambda-Matrices and Vibrating Systems, Dover Publications, 2002.
7
[6] J.M. Gracia, Nearest matrix with two prescribed eigenvalues, Linear Algebra Appl., 401 (2005), 277-294.
8
[7] R.A. Lippert, Fixing two eigenvalues by a minimal perturbation, Linear Algebra Appl., 406 (2005) 177-200.
9
[8] A.N. Malyshev, A formula for the 2-norm distance from a matrix to the set of matrices with a multiple eigen-
10
value, Numer. Math., 83 (1999) 443-454.
11
[9] A.S. Markus, Introduction to the Spectral Theory of Polynomial Operator Pencils, Amer. Math., Society, Provi-
12
dence, RI, Translations of Mathematical Monographs, Vol. 71, 1988.
13
[10] J. Nocedal, S.J. Wright, Numerical Optimization, second edition, Springer Series in Operation Research and
14
Financial Engineering, 2006.
15
[11] N. Papathanasiou, P. Psarrakos, The distance from a matrix polynomial to matrix polynomials with a prescribed
16
multiple eigenvalue, Linear Algebra Appl., 429 (2008), 1453-1477.
17
[12] A. Ruhe, Properties of a matrix with a very ill-conditioned eigenproblem, Numer. Math., 15 (1970), 57–60.
18
[13] J.H. Wilkinson, The Algebraic Eigenvalue Problem, Claredon Press, Oxford, 1965.
19
[14] J.H. Wilkinson, Note on matrices with a very ill-conditioned eigenproblem, Numer. Math., 19 (1972), 175–178.
20
[15] J.H. Wilkinson, On neighbouring matrices with quadratic elementary divisors, Numer. Math., 44 (1984), 1–21.
21
[16] J.H. Wilkinson, Sensitivity of eigenvalues, Util. Math., 25 (1984), 5–76.
22
[17] J.H. Wilkinson, Sensitivity of eigenvalues II, Util. Math., 30 (1986), 243–286.
23
ORIGINAL_ARTICLE
G-dual function-valued frames in L2(0,∞)
>In this paper, g-dual function-valued frames in L2(0;1) are in- troduced. We can achieve more reconstruction formulas to ob- tain signals in L2(0;1) by applying g-dual function-valued frames in L2(0;1).
http://wala.vru.ac.ir/article_14268_9b2b81f3b7b0c0ef1d67b31101cdd174.pdf
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39
47
g-dual frame
function-valued frame
M. A.
Hasankhanifard
true
1
Vali-e-Asr university of Rafsanjan
Vali-e-Asr university of Rafsanjan
Vali-e-Asr university of Rafsanjan
LEAD_AUTHOR
M. A.
Dehghan
true
2
Vali-e-Asr university of Rafsanjan
Vali-e-Asr university of Rafsanjan
Vali-e-Asr university of Rafsanjan
AUTHOR
[1] P.G. Casazza, The Art of Frame Theory, Taiwanese J. Math, 4 (2000), 129-201.
1
[2] P. G. Casazza, and G. Kutyniok, Frames of Subspaces, Contemporary Math, 345 (2004), 87-114.
2
[3] P. G. Casazza and M. C. Lammers, Bracket Products for Weyl-Heisenberg Frames, In: Advances in Gabor
3
Analysis (eds) Feichtinger H G and Strohmer T (2003 ) (Boston-MA. Birkh¨ auser).
4
[4] O. Christensen, An Introduction to Frames and Riesz Bases, Birkh¨auser, Boston, Basel, Berlin, 2002.
5
[5] O. Christensen and Y. Eldar, Oblique dual frames and shift-invariant spaces, Appl. Comp. Harm. Anal., 17 (2004)
6
[6] O. Christensen and R. S. Laugesen, Approximately dual frames in Hilbert spaces and application to Gabor
7
frames, Sampl. Theory Signal Image Process. 9 (2011), 77-90.
8
[7] M. A. Dehghan and M. A. Hasankhani Fard, G-dual frames in Hilbert spaces, U.P.B. Sci. Bull., Series A, 75(1)
9
(2013), 129-140.
10
[8] M. A. Hasankhani Fard and M. A. Dehghan, A new function-valued inner product and corresponding function-
11
valued frames in L2(0;1), Linear Multilinear Algebra, (Published online: 01 Jul 2013).
12
[9] A. A. Hemmat and J. P. Gabardo, Properties of oblique dual frames in shift-invariant systems, J. Math. Anal.
13
Appl., 356 (2009) 346-354.
14
10] S. Li and H. Ogawa, Pseudo duals of frames with applications, Appl. Comput. Harmon. Anal., 11 (2001) 289-
15
11] R. Young, An Introduction to Nonharmonic Fourier Series, Academic Press, New York, 1980.
16
ORIGINAL_ARTICLE
Schur multiplier norm of product of matrices
>For A ∈ M n, the Schur multiplier of A is defined as S A(X) = A ◦ X for all X ∈ M n and the spectral norm of S A can be state as ∥S A∥ = supX,0 ∥A ∥X ◦X ∥ ∥. The other norm on S A can be defined as ∥S A∥ω = supX,0 ω(ω S( AX (X ) )) = supX,0 ωω (A (X ◦X ) ), where ω(A) stands for the numerical radius of A. In this paper, we focus on the relation between the norm of Schur multiplier of product of matrices and the product of norm of those matrices. This relation is proved for Schur product and geometric product and some applications are given. Also we show that there is no such relation for operator product of matrices. Furthermore, for positive definite matrices A and B with ∥S A∥ω ⩽ 1 and ∥S B∥ω ⩽ 1, we show that A♯B = n(I − Z)1/2C(I + Z)1/2, for some contraction C and Hermitian contraction Z.
http://wala.vru.ac.ir/article_14269_e9a1dc6d7c67b98d454cd7225318629e.pdf
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54
Schur multiplier
Schur product
Geometric product
Positive semideﬁnite
matrix
Numerical radius
M.
Khosravi
true
1
Shahid Bahonar university of Kerman
Shahid Bahonar university of Kerman
Shahid Bahonar university of Kerman
LEAD_AUTHOR
A.
Sheikhhosseini
true
2
Shahid Bahonar university of Kerman
Shahid Bahonar university of Kerman
Shahid Bahonar university of Kerman
AUTHOR
[1] T. Ando, On the structure of operators with numerical radius one, Acta Sci. Math., (Szeged) 34 (1973), 11–15.
1
[2] T. Ando and K. Okubo, Induced norms of the Schur multiplier operator, Linear Algebra Appl., 147 (1991),
2
181–199.
3
[3] J.R. Angelos, C.C. Cowen and S.K. Narayan, Triangular truncation and finding the norm of a Hadamard multiplier, Linear Algebra Appl., 170 (1992), 181–199.
4
[4] K.M.R. Audenaert, Schur multiplier norms for Loewner matrices, Linear Algebra Appl., 439 (2013), 2598–
5
[5] R. Bhatia, Positive Definite Matrices, Princeton University Press, Princeton, 2007.
6
[6] P. Chaisuriya, A C∗-algebra on Schur algebras, Bull. Malays. Math. Sci. Soc., 34(2) (2011), 189-200.
7
[7] P. Chaisuriya and S.-C. Ong, On Schur Multiplier Norm and Unitaries, Southeast Asian Bull. Math., 26 (2003),
8
no. 6, 889–898.
9
[8] C.K. Fong, H. Radjavi and P. Rosenthal, norms for matrices and operators, J. Operator Theory, 18 (1987),
10
99–113.
11
[9] I.C. Gohberg and M.G. Krein, Theory and applications of Volterra operators in Hilbert space, English translation, AMS, Providence, 1970.
12
[10] R.A. Horn and C.R. Johnson, Matrix Analysis, Second ed. Cambridge university press, 2012.
13
[11] A. Katavolos and V. Paulsen, On the ranges of bimodule projections, Canad. Math. Bull., 48(1)(2005), 91–111.
14
[12] L. Livshits, Generalized Schur Products for Matrices with Operator Entries, ProQuest LLC, Ann Arbor, MI,
15
[13] L. Livschits, A note on 0-1 Schur multipliers, Linear Algebra Appl. 22 (1995), 15–22.
16
[14] R. Mathias, An arithmetic-geometric-harmonic mean inequality involving Hadamard products, Linear Algebra
17
Appl. 184 (1993), 71–78.
18
[15] S.-C. Ong, On the Schur multiplier of matrices, Linear Algebra Appl. 56 (1984), 45–55.
19
ORIGINAL_ARTICLE
Ultra Bessel sequences in direct sums of Hilbert spaces
>In this paper, we establish some new results in ultra Bessel sequences and ultra Bessel sequences of subspaces. Also, we investigate ultra Bessel sequences in direct sums of Hilbert spaces. Specially, we show that {( fi, gi)}∞ i=1 is a an ultra Bessel sequence for Hilbert space H ⊕ K if and only if { fi}∞ i=1 and {gi}∞ i=1 are ultra Bessel sequences for Hilbert spaces H and K, respectively.
http://wala.vru.ac.ir/article_14270_489abf58eb663e969ea22c4d90360acb.pdf
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55
64
Frame of subspaces
Ultra Bessel sequence
M. R.
Abdollahpour
true
1
University
of Mohaghegh Ardabili
University
of Mohaghegh Ardabili
University
of Mohaghegh Ardabili
LEAD_AUTHOR
A.
Rahimi
true
2
University of Maragheh
University of Maragheh
University of Maragheh
AUTHOR
[1] Abdollahpour, M. R. and Najati, A. Ultra g-Bessel sequences in Hilbert spaces, Kyungpook Math. J., 54 (2014),
1
[2] Abdollahpour,M. R., Shekari, A., Park, C., and Shin, D. Y. Ultra Bessel sequences of subspaces in Hilbert spaces,
2
J. Compt. Anal Appl. 19 (5) (2015), 874-882.
3
[3] Abdollahpour, M. R. and Shekari, A., Frameness bound for frame of subspaces, Sahand Communications in
4
Mathematical Analysis, 1 (1) (2014), 1-8.
5
[4] Casazza, P. G. and Kutyniok, G. Frames of subspaces, Contempt. Math. Amer. Math. Soc. providence., 345
6
(2004), 87-113.
7
[5] Christencen, O. An introduction to frames and Riesz bases, Birkhauser, Boston, 2003.
8
[6] Faroughi, M. H. and Najati, A. Ultra Bessel sequences in Hilbert Spaces, Southeast Asian Bull. Math., 32 (2008),
9
[7] Khosravi, A. and Asgari, M. S. Frames of subspaces an aproximation of the inverse frame operator, Houston J.
10
Math., 33 (3) (2007), 907-920.
11
ORIGINAL_ARTICLE
Some relations between ε-directional derivative and ε-generalized weak subdifferential
>In this paper, we study ε-generalized weak subdifferential for vector valued functions defined on a real ordered topological vector space X. We give various characterizations of ε-generalized weak subdifferential for this class of functions. It is well known that if the function f : X → R is subdifferentiable at x0 ∈ X, then f has a global minimizer at x0 if and only if 0 ∈ ∂ f(x0). We show that a similar result can be obtained for ε-generalized weak subdifferential. Finally, we investigate some relations between ε-directional derivative and ε-generalized weak subdifferential. In fact, in the classical subdifferential theory, it is well known that if the function f : X → R is subdifferentiable at x0 ∈ X and it has directional derivative at x0 in the direction u ∈ X, then the relation f ′(x0, u) ≥ ⟨u, x∗⟩, ∀ x∗ ∈ ∂ f(x0) is satisfied. We prove that a similar result can be obtained for ε- generalized weak subdifferential.
http://wala.vru.ac.ir/article_14591_7255b9cf0db6154ec39af397e9141d48.pdf
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65
80
Non-convex optimization
"-directional derivative
A.
Mohebi
true
1
Shahid Bahonar university of Kerman
Shahid Bahonar university of Kerman
Shahid Bahonar university of Kerman
AUTHOR
H.
Mohebi
true
2
Shahid Bahonar university of Kerman
Shahid Bahonar university of Kerman
Shahid Bahonar university of Kerman
AUTHOR
[1] A.Y. Azimov and R.N. Gasmiov, On weak conjugacy, weak subdifferentials and duality with zero gap in nonconvex optimization, Int. J. Appl. Math., 1(1999), 171-192.
1
[2] A.Y. Azimov and R.N. Gasmiov, Stability and duality of nonconvex problems via augmented Lagrangian, Cybernet. Syst. Anal., 38(2002), 412-421.
2
[3] J.M. Borwein, Continuity and differentiability properties of convex operators, Proc. London Math. Soc.,
3
44(1982), 420-444.
4
[4] R.N. Gasimov, Augmented Lagrangian duality and nondifferentiable optimization methods in nonconvex programing, J. Global Optim., 24(2002), 187-203.
5
[5] Guang-ya Chen, Xuexiang Huang and Xiaogi Yang, Vector optimization: Set-Valued and Variational Analysis,
6
Springer, Berlin, 2005.
7
[6] J. Jahn, Vector optimization, Springer, Berlin, 2004.
8
[7] Y. Kuc ¨ uk, L. Ataserer and M. K ¨ uc ¨ uk, ¨ Generalized weak subdifferentials, Optimization, 60(5)(2011), 537-552.
9
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