Weaving frames in separable Hilbert spaces have been recently introduced by Bemrose et al. to deal with some problems in distributed signal processing and wireless sensor networks. Likewise weaving K -frames have been proved to be useful during signal reconstructions from the range of a bounded linear operator K. In this paper, we study the notion of weaving and its connection to the duality of K-frames and construct several pairs of woven K-frames. Also, we find a unique biorthogonal sequence for every K-Riesz basis and obtain a $K^*$-frame which is woven by its canonical dual. Moreover, we describe the excess for K-frames and prove that any two woven K-frames in a separable Hilbert space have the same excess. Finally, we introduce the necessary and sufficient condition under which a K-frame and its image under an invertible operator have the same excess.
[1] E. Agheshteh Moghadam and A. Arefijamaal, On Excesses and Duality in Woven
Frames, Bull. Malays. Math. Sci. Soc. 44 (5), 3361–3375, 2021.
[2] A. Aldroubi, Portraits of Frames, Proc. Amer. Math. Soc. 123 (1), 1661–1668, 1995.
[3] F. Arabyani Neyshaburi and A. Arefijamaal, Manufactoring Pairs of Woven Frames
Applying Duality Principle on Hilbert Spaces, Bull. Malays. Math. Sci. Soc. 44 (1),
147–161, 2020.
[4] F. Arabyani Neyshaburi and A. Arefijamaal, Some construction of K-frames and their
duals, Rocky Mt. J. Math. 47 (6), 1749–1764, 2017.
[5] A. Arefijamaal and E. Zekaee, Image Processing by Alternate Dual Gabor Frames,
Bull. Iranian Math. Soc. 42 (6), 1305–1314, 2016.
[6] A. Arefijamaal and E. Zekaee, Signal Processing by Alternate Dual Gabor Frames,
Appl. Comput. Harmon. Anal. 35, 535–540, 2013.
[7] A. Bhandari, D. Borah and S. Mukherjee, Characterizations of Weaving K-frames,
Proc. Japan Acad. Ser. A Math. Sci. 96 (5), 39–43, 2020.
[8] D. Bakic, T. Beric, On excesses of frames, Glas. Mat. Ser. III 50 (2), 415–427, 2015.
[9] P. Balazs and K. Gröchenig. A guide to localized frames and applications to galerkinlike
representations of operators. In I. Pesenson, H. Mhaskar, A. Mayeli, Q. T. L.
Gia, and D.-X. Zhou, editors, Novel methods in harmonic analysis with applications to numerical analysis and data processing, Applied and Numerical Harmonic Analysis
series (ANHA). Birkhauser/Springer, 2017.
[10] P. Balazs, M. Shamsabadi, A. Arefijamaal, and A. Rahimi, U-cross Gram matrices
and their invertibility, J. Math. Anal. Appl. 476 (2), 367–390, 2019.
[11] T. Bemrose, P. G. Casazza, K. Grochenig,M. C. Lammers and R. G Lynch, Weaving
frames, Oper. Matrices 10 (4), 1093–1116, 2016.
[12] H. Boleskel,F. Hlawatsch and H. G. Feichtinger , Frame Theoretic Analysis of Oversampled
Filter Banks, IEEE Trans. Signal Process 46, 3256–3268, 1998.
[13] P. G. Casazza and G. Kutyniok, Frames of Subspaces, Contempt. Math. 345, 87–113,
2004.
[14] P. G. Casazza, The art of Frame Theory, Taiwanese J. Math. 4 (2), 129–202, 2000.
[15] O. Christensen, Frames and Bases: An Introductory Course, Birkhäuser, Boston,
2008.
[16] N. Cotfas and J. P. Gazeau. Finite tight frames and some applications, J. Phys. A:
Math. Theor. 43 (19), 193001, 2010.
[17] Deepshikha and Lalit K. Vashisht, Weaving K-Frames in Hilbert spaces, Results
Math. 27, 73–81, 2018 .
[18] R. Duffin and A. Schaeffer, A Class of Nonharmonic Fourier Series, Trans. Amer.
Math. Soc. 72, 341–366, 1952.
[19] Y.C. Eldar, O. Christensen, A characterization of oblique dual frame Pairs,
EURASIP J. Adv. Sig. Proc. 1–11, 2006.
[20] S. Garg and L.K. Vashisht, Weaving K-Fusion Frames in Hilbert Spaces, Ganita 67
(1), 41–52, 2017.
[21] L. Gˇavruta, Frames for Operators, Appl. Comp. Harmon. Anal. Appl. 32, 139–144,
2012.
[22] J.R. Holub, Pre-Frames Operators, Besselian Frames and near-Riesz Bases in Hilbert
Spaces, Amer. Math. Soc. 122, 779–785, 1994.
[1] E. Agheshteh Moghadam and A. Arefijamaal, On Excesses and Duality in Woven
Frames, Bull. Malays. Math. Sci. Soc. 44 (5), 3361–3375, 2021.
[2] A. Aldroubi, Portraits of Frames, Proc. Amer. Math. Soc. 123 (1), 1661–1668, 1995.
[3] F. Arabyani Neyshaburi and A. Arefijamaal, Manufactoring Pairs of Woven Frames
Applying Duality Principle on Hilbert Spaces, Bull. Malays. Math. Sci. Soc. 44 (1),
147–161, 2020.
[4] F. Arabyani Neyshaburi and A. Arefijamaal, Some construction of K-frames and their
duals, Rocky Mt. J. Math. 47 (6), 1749–1764, 2017.
[5] A. Arefijamaal and E. Zekaee, Image Processing by Alternate Dual Gabor Frames,
Bull. Iranian Math. Soc. 42 (6), 1305–1314, 2016.
[6] A. Arefijamaal and E. Zekaee, Signal Processing by Alternate Dual Gabor Frames,
Appl. Comput. Harmon. Anal. 35, 535–540, 2013.
[7] A. Bhandari, D. Borah and S. Mukherjee, Characterizations of Weaving K-frames,
Proc. Japan Acad. Ser. A Math. Sci. 96 (5), 39–43, 2020.
[8] D. Bakic, T. Beric, On excesses of frames, Glas. Mat. Ser. III 50 (2), 415–427, 2015.
[9] P. Balazs and K. Gröchenig. A guide to localized frames and applications to galerkinlike
representations of operators. In I. Pesenson, H. Mhaskar, A. Mayeli, Q. T. L.
Gia, and D.-X. Zhou, editors, Novel methods in harmonic analysis with applications to numerical analysis and data processing, Applied and Numerical Harmonic Analysis
series (ANHA). Birkhauser/Springer, 2017.
[10] P. Balazs, M. Shamsabadi, A. Arefijamaal, and A. Rahimi, U-cross Gram matrices
and their invertibility, J. Math. Anal. Appl. 476 (2), 367–390, 2019.
[11] T. Bemrose, P. G. Casazza, K. Grochenig,M. C. Lammers and R. G Lynch, Weaving
frames, Oper. Matrices 10 (4), 1093–1116, 2016.
[12] H. Boleskel,F. Hlawatsch and H. G. Feichtinger , Frame Theoretic Analysis of Oversampled
Filter Banks, IEEE Trans. Signal Process 46, 3256–3268, 1998.
[13] P. G. Casazza and G. Kutyniok, Frames of Subspaces, Contempt. Math. 345, 87–113,
2004.
[14] P. G. Casazza, The art of Frame Theory, Taiwanese J. Math. 4 (2), 129–202, 2000.
[15] O. Christensen, Frames and Bases: An Introductory Course, Birkhäuser, Boston,
2008.
[16] N. Cotfas and J. P. Gazeau. Finite tight frames and some applications, J. Phys. A:
Math. Theor. 43 (19), 193001, 2010.
[17] Deepshikha and Lalit K. Vashisht, Weaving K-Frames in Hilbert spaces, Results
Math. 27, 73–81, 2018 .
[18] R. Duffin and A. Schaeffer, A Class of Nonharmonic Fourier Series, Trans. Amer.
Math. Soc. 72, 341–366, 1952.
[19] Y.C. Eldar, O. Christensen, A characterization of oblique dual frame Pairs,
EURASIP J. Adv. Sig. Proc. 1–11, 2006.
[20] S. Garg and L.K. Vashisht, Weaving K-Fusion Frames in Hilbert Spaces, Ganita 67
(1), 41–52, 2017.
[21] L. Gˇavruta, Frames for Operators, Appl. Comp. Harmon. Anal. Appl. 32, 139–144,
2012.
[22] J.R. Holub, Pre-Frames Operators, Besselian Frames and near-Riesz Bases in Hilbert
Spaces, Amer. Math. Soc. 122, 779–785, 1994.
Agheshteh Moghaddam, E., & Arefijamaal, A. A. (2024). New aspects of weaving K-frames: the excess and duality. Hacettepe Journal of Mathematics and Statistics, 53(3), 652-666. https://doi.org/10.15672/hujms.1008448
AMA
Agheshteh Moghaddam E, Arefijamaal AA. New aspects of weaving K-frames: the excess and duality. Hacettepe Journal of Mathematics and Statistics. June 2024;53(3):652-666. doi:10.15672/hujms.1008448
Chicago
Agheshteh Moghaddam, Elahe, and Ali Akbar Arefijamaal. “New Aspects of Weaving K-Frames: The Excess and Duality”. Hacettepe Journal of Mathematics and Statistics 53, no. 3 (June 2024): 652-66. https://doi.org/10.15672/hujms.1008448.
EndNote
Agheshteh Moghaddam E, Arefijamaal AA (June 1, 2024) New aspects of weaving K-frames: the excess and duality. Hacettepe Journal of Mathematics and Statistics 53 3 652–666.
IEEE
E. Agheshteh Moghaddam and A. A. Arefijamaal, “New aspects of weaving K-frames: the excess and duality”, Hacettepe Journal of Mathematics and Statistics, vol. 53, no. 3, pp. 652–666, 2024, doi: 10.15672/hujms.1008448.
ISNAD
Agheshteh Moghaddam, Elahe - Arefijamaal, Ali Akbar. “New Aspects of Weaving K-Frames: The Excess and Duality”. Hacettepe Journal of Mathematics and Statistics 53/3 (June 2024), 652-666. https://doi.org/10.15672/hujms.1008448.
JAMA
Agheshteh Moghaddam E, Arefijamaal AA. New aspects of weaving K-frames: the excess and duality. Hacettepe Journal of Mathematics and Statistics. 2024;53:652–666.
MLA
Agheshteh Moghaddam, Elahe and Ali Akbar Arefijamaal. “New Aspects of Weaving K-Frames: The Excess and Duality”. Hacettepe Journal of Mathematics and Statistics, vol. 53, no. 3, 2024, pp. 652-66, doi:10.15672/hujms.1008448.
Vancouver
Agheshteh Moghaddam E, Arefijamaal AA. New aspects of weaving K-frames: the excess and duality. Hacettepe Journal of Mathematics and Statistics. 2024;53(3):652-66.