Abstract
The matrix representation of operators in Hilbert spaces is a useful tool in applications. It is important to present the matrix representation by sequences other than orthonormal bases. In this paper, we extend the matrix representation of operators using g-frames and investigate their invertibility and stability.
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Abdollahi, A.; Rahimi, A.: Some results on \(g\)-frames in Hilbert spaces. Turk. J. Math. 34, 695–704 (2010)
Ali, S.T.; Antoine, J.-P.; Gazeau, J.-P.: Continous frames in Hilbert spaces. Ann. Phys. 222, 1–37 (1993)
Ali, S.T.; Antoine, J.-P.; Gazeau, J.-P.: Coherent States. Wavelets and Their Generalization. Graduate Texts in Contemporary Physics. Springer, New York (2000)
Arefijamaal, A.A.; Ghasemi, S.: On characterization and stability of alternate dual of g-frames. Turk. J. Math. 37, 71–79 (2013)
Arefijamaal, A.; Zekaee, E.: Image processing by alternate dual Gabor frames. Bull. Iran. Math. Soc. 42(6), 1305–1314 (2016)
Arefijamaal, A.; Zekaee, E.: Signal processing by alternate dual Gabor frames. Appl. Comput. Harmon. Anal. 35, 535–540 (2013)
Balazs, P.: Matrix-representation of operators using frames. Sampl. Theory Signal Image Process. (STSIP) 7(1), 39–54 (2008)
Balazs, P.; Kreuzer, W.; Waubke, H.: A stochastic 2d-model for calculating vibrations in liquids and soils. J. Comput. Acoust. 15(3), 271–283 (2007)
Benedetto, J.; Powell, A.; Yilmaz, O.: Sigm-delta quantization and finite frames. IEEE Trans. Inf. Theory. 52, 1990–2005 (2006)
Bodmannand, B.G.; Paulsen, V.I.: Frames, graphs and erasures. Linear Algebra Appl. 404, 118–146 (2005)
Bolcskel, H.; Hlawatsch, F.; Feichtinger, H.G.: Frame-theoretic analysis of oversampled filter banks. IEEE Trans. Signal Process. 46, 3256–3268 (1995)
Candes, E.J.; Donoho, D.L.: New tight frames of curvelets and optimal representations of objects with piecewise \(C^2\) singularities. Commun. Pure Appl. Anal. 56, 216–266 (2004)
Casazza, P.G.; Kutyniok, G.; Li, S.: Fusion frames and distributed processing. Appl. Comput. Harmon. Anal. 25(1), 132–144 (2008)
Christensen, O.: Frames and Bases: An Introductory Course. Birkhäuser, Boston (2008)
Christensen, O.: Frames and pseudo-inverses. J. Math. Anal. Appl. 195(2), 401–414 (1995)
Conway, J.B.: A Course in Functional Analysis, 2nd edn. Graduate Texts in Mathematics. Springer, New York (1990)
Daubechies, I.: The wavelet transform, time frequency localization and signal analysis. IEEE Trans. Inf. Theory 36(5), 961–1005 (1990)
Daubechies, I.; Grossmann, A.; Meyer, Y.: Painless nonorthogonal expansions. J. Math. Phys. 27, 1271–1283 (1986)
Duffin, R.; Schaeffer, A.: A class of nonharmonic Fourier series. Trans. Am. Math. Soc. 72, 341–366 (1952)
Feichtinger, H.G.; Grochenig, K.: Irregular sampling theorems and series expansion of band-limited functions. Math. Anal. Appl. 167, 530–556 (1992)
Feichtinger, H.G.; Strohmer, T.: Gabor Analysis and Algorithms—Theory and Applications. Birkhauser, Boston (1998)
Fillmore, P.A.; Williams, J.P.: On operator ranges. Adv. Math. 7, 254–281 (1971)
Futamura, F.: Frame diagonalization of matrices. Linear Algebra Appl. 436(9), 3201–3214 (2012)
Gohberg, I.; Goldberg, S.; Kaashoek, M.: Basic Classes of Linear Operators. Birkhäuser, Basel (2003)
Jin, G.; Chen, A.: Some basic properties of block operator matrices. arXiv:1403.7732
Kirkup, S.M.; Wadsworth, M.: Solution of inverse diffusion problems by operator-splitting methods. Appl. Math. Model. 26, 1003–1018 (2002)
Mitchell, A.R.; Griffiths, D.F.: The Finite Difference Method in Partial Differential Equations. Wiley, New York (1980)
Mallat, S.: A Wavelet Tour of Signal Processing, 2nd edn. Academic Press, Cambridge (1999)
Najati, A.; Faroughi, M.H.; Rahimi, A.: \(g\)-frames and stability of \(g\)-frames in Hilbert spaces. Methods Funct. Anal. Topol. 4(3), 271–286 (2008)
Radol, K.: Matrices related to some fock space operators. Opuscula Math. 2, 289–297 (2011)
Shamsabadi, M., Arefijamaal, A., Balazs, P., Rahimi, A.: \(U\)-cross Gram matrices and their associated reconstructions. arXiv:1804.00203
Sowa, A.: Encoding spatial data into quantum observables. arXiv:1609.01712
Sun, W.: \(G\)-frames and \(g\)-Riesz bases. J. Math. Anal. Appl. 322, 437–452 (2006)
Sun, W.: Stability of \(g\)-frames. J. Math. Anal. Appl. 326(2), 858–868 (2007)
Zhou, P.: Numerical Analysis of Electromagnetic Fields. Springer, New York (1993)
Zhu, Y.C.: Characterizations of \(g\)-frames and \(g\)-bases in Hilbert spaces. Acta Math. Sin. 24, 1727–1736 (2008)
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The authors are thankful to P. Balazs for his valuable comments. They also want to thank all anonymous reviewers concerned with this paper.
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Arefijamaal, A.A., Shamsabadi, M. O-Cross Gram matrices with respect to \(\varvec{g}\)-frames. Arab. J. Math. 9, 259–269 (2020). https://doi.org/10.1007/s40065-019-0246-8
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DOI: https://doi.org/10.1007/s40065-019-0246-8