Year 2023,
, 176 - 183, 15.09.2023
Rosario Corso
Project Number
FSE - REACT EU, PON Ricerca e Innovazione 2014-2020
References
- R. Balan, P.G. Casazza, C. Heil and Z. Landau: Density, overcompleteness, and localization of frames I. Theory, J. Fourier Anal. Appl., 12 (2006), 105–143.
- P. Balazs: Basic definition and properties of Bessel multipliers, J. Math. Anal. Appl., 325 (1) (2007), 571–585.
- P. Balazs: Hilbert-Schmidt operators and frames-classification, best approximation by multipliers and algorithms, Int. J. Wavelets Multiresolut. Inf. Process., 6 (2) (2008), 315–330.
- P. Balazs, B. Laback, G. Eckel and W.A. Deutsch: Time-frequency sparsity by removing perceptually irrelevant components using a simple model of simultaneous masking, IEEE Transactions on Audio, Speech, and Language Processing, 18 (1) (2010), 34–49.
- P. Balazs, D. T. Stoeva: Representation of the inverse of a frame multiplier, J. Math. Anal. Appl., 422 (2) (2015), 981–994.
- O. Christensen: An Introduction to Frames and Riesz Bases, second expanded edition, Birkhäuser, Boston, (2016).
- J. Conway: A Course in Functional Analysis, Graduate Texts in Mathematics. 96 (2nd ed.), New York: Springer-Verlag, (1990).
- E. Cordero, K. Gröchenig: Localization of frames II, Appl. Comput. Harmon. Anal., 17 (2004), 29–47.
- R. Corso: On some dual frames multipliers with at most countable spectra, Ann. Mat. Pura Appl., 201 (4) (2022), 1705–1716.
- R. Corso: Localization of the spectra of dual frames multipliers, Constr. Math. Anal., 5 (4) (2022), 238–245.
- R. Corso, F. Tschinke: Some notes about distribution frame multipliers, in: Landscapes of Time-Frequency Analysis, vol. 2, P. Boggiatto, T. Bruno, E. Cordero, H.G. Feichtinger, F. Nicola, A. Oliaro, A. Tabacco, M. Vallarino (Ed.), Applied and Numerical Harmonic Analysis Series, Springer, (2020).
- H. G. Feichtinger, K. Nowak: A first survey of Gabor multipliers, in: Advances in Gabor analysis, H. G. Feichtinger and T. Strohmer (Ed.), Boston Birkhäuser, Applied and Numerical Harmonic Analysis, (2003).
- G. B. Folland: A Course in Abstract Harmonic Analysis, CRC Press, Boca, Raton, (1995).
- J.-P. Gazeau: Coherent States in Quantum Physics, Weinheim: Wiley, (2009).
- K. Gröchenig: Foundations of Time-Frequency Analysis, Birkhäauser, Boston, (2000).
- K. Gröchenig: Localization of frames, Banach frames, and the invertibility of the frame operator, J. Fourier Anal. Appl., 10 (2004), 105–132.
- K. Gröchenig, M. Fornasier: Intrinsic localization of frames, Constr. Approx., 22 (2005), 395–415.
- E. Hernández, H. Šiki´c, G. Weiss and E. N. Wilson: Cyclic subspaces for unitary representations of LCA groups; generalized Zak transform, Colloq. Math., 118 (1) (2010), 313–332.
- H. Javanshiri, M. Abolghasemi, A.A. Arefijamaal: The essence of invertible frame multipliers in scalability, Adv. Comput. Math., 48 (2022), ARTICLE ID: 19.
- T. Kato: Perturbation Theory for Linear Operators, Springer, Berlin, (1966).
- G. Matz, F. Hlawatsch: Linear time-frequency filters: On-line algorithms and applications, in: A. Papandreou-
Suppappola (Ed.), Application in Time-Frequency Signal Processing, CRC Press, Boca Raton, FL, (2002).
- R. Levie, H. Avron: Randomized Signal Processing with Continuous Frames, J. Fourier Anal. Appl., 28 (2022), ARTICLE ID: 5.
- K. Schmüdgen: Unbounded Self-adjoint Operators on Hilbert Space, Springer, Dordrecht, (2012).
- D. T. Stoeva, P. Balazs: Invertibility of multipliers, Appl. Comput. Harmon. Anal., 33 (2) (2012), 292–299.
- D. T. Stoeva, P. Balazs: On the dual frame induced by an invertible frame multiplier, Sampl. Theory Signal and Image Proc., 15 (2016), 119–130.
Estimate of the spectral radii of Bessel multipliers and consequences
Year 2023,
, 176 - 183, 15.09.2023
Rosario Corso
Abstract
Bessel multipliers are operators defined from two Bessel sequences of elements of a Hilbert space and a complex sequence, and have frame multipliers as particular cases. In this paper an estimate of the spectral radius of a Bessel multiplier is provided involving the cross Gram operator of the two sequences. As an upshot, it is possible to individuate some regions of the complex plane where the spectrum of a multiplier of dual frames is contained.
Supporting Institution
European Union
Project Number
FSE - REACT EU, PON Ricerca e Innovazione 2014-2020
Thanks
European Union, GNAMPA-INdAM
References
- R. Balan, P.G. Casazza, C. Heil and Z. Landau: Density, overcompleteness, and localization of frames I. Theory, J. Fourier Anal. Appl., 12 (2006), 105–143.
- P. Balazs: Basic definition and properties of Bessel multipliers, J. Math. Anal. Appl., 325 (1) (2007), 571–585.
- P. Balazs: Hilbert-Schmidt operators and frames-classification, best approximation by multipliers and algorithms, Int. J. Wavelets Multiresolut. Inf. Process., 6 (2) (2008), 315–330.
- P. Balazs, B. Laback, G. Eckel and W.A. Deutsch: Time-frequency sparsity by removing perceptually irrelevant components using a simple model of simultaneous masking, IEEE Transactions on Audio, Speech, and Language Processing, 18 (1) (2010), 34–49.
- P. Balazs, D. T. Stoeva: Representation of the inverse of a frame multiplier, J. Math. Anal. Appl., 422 (2) (2015), 981–994.
- O. Christensen: An Introduction to Frames and Riesz Bases, second expanded edition, Birkhäuser, Boston, (2016).
- J. Conway: A Course in Functional Analysis, Graduate Texts in Mathematics. 96 (2nd ed.), New York: Springer-Verlag, (1990).
- E. Cordero, K. Gröchenig: Localization of frames II, Appl. Comput. Harmon. Anal., 17 (2004), 29–47.
- R. Corso: On some dual frames multipliers with at most countable spectra, Ann. Mat. Pura Appl., 201 (4) (2022), 1705–1716.
- R. Corso: Localization of the spectra of dual frames multipliers, Constr. Math. Anal., 5 (4) (2022), 238–245.
- R. Corso, F. Tschinke: Some notes about distribution frame multipliers, in: Landscapes of Time-Frequency Analysis, vol. 2, P. Boggiatto, T. Bruno, E. Cordero, H.G. Feichtinger, F. Nicola, A. Oliaro, A. Tabacco, M. Vallarino (Ed.), Applied and Numerical Harmonic Analysis Series, Springer, (2020).
- H. G. Feichtinger, K. Nowak: A first survey of Gabor multipliers, in: Advances in Gabor analysis, H. G. Feichtinger and T. Strohmer (Ed.), Boston Birkhäuser, Applied and Numerical Harmonic Analysis, (2003).
- G. B. Folland: A Course in Abstract Harmonic Analysis, CRC Press, Boca, Raton, (1995).
- J.-P. Gazeau: Coherent States in Quantum Physics, Weinheim: Wiley, (2009).
- K. Gröchenig: Foundations of Time-Frequency Analysis, Birkhäauser, Boston, (2000).
- K. Gröchenig: Localization of frames, Banach frames, and the invertibility of the frame operator, J. Fourier Anal. Appl., 10 (2004), 105–132.
- K. Gröchenig, M. Fornasier: Intrinsic localization of frames, Constr. Approx., 22 (2005), 395–415.
- E. Hernández, H. Šiki´c, G. Weiss and E. N. Wilson: Cyclic subspaces for unitary representations of LCA groups; generalized Zak transform, Colloq. Math., 118 (1) (2010), 313–332.
- H. Javanshiri, M. Abolghasemi, A.A. Arefijamaal: The essence of invertible frame multipliers in scalability, Adv. Comput. Math., 48 (2022), ARTICLE ID: 19.
- T. Kato: Perturbation Theory for Linear Operators, Springer, Berlin, (1966).
- G. Matz, F. Hlawatsch: Linear time-frequency filters: On-line algorithms and applications, in: A. Papandreou-
Suppappola (Ed.), Application in Time-Frequency Signal Processing, CRC Press, Boca Raton, FL, (2002).
- R. Levie, H. Avron: Randomized Signal Processing with Continuous Frames, J. Fourier Anal. Appl., 28 (2022), ARTICLE ID: 5.
- K. Schmüdgen: Unbounded Self-adjoint Operators on Hilbert Space, Springer, Dordrecht, (2012).
- D. T. Stoeva, P. Balazs: Invertibility of multipliers, Appl. Comput. Harmon. Anal., 33 (2) (2012), 292–299.
- D. T. Stoeva, P. Balazs: On the dual frame induced by an invertible frame multiplier, Sampl. Theory Signal and Image Proc., 15 (2016), 119–130.