Abstract
In this short article, we establish some new inequalities for Parseval generalized continuous frames and give another proof of existing two inequalities for Parseval generalized continuous frames.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
R.J. Duffin, A.C. Schaeffer, A class of nonharmonic Fourier series. Trans. Amer. Math. Soc. 72, 341–366 (1952)
E.J. Candés, D.L. Donoho, New tight frames of curvelets and optimal representations of objects with piecewise \(\mathcal {C}^2\) singularities. Comm. Pure Appl. Math. 56, 216–266 (2004)
E.J. Candés, Harmonic analysis and neural networks. Appl. Comput. Harmon. Anal. 6, 197–218 (1999)
T. Strohmer, R. W. Heath Jr, Grassmannian frames with applications to coding and communication. Appl. Comput. Harmon. Anal. 14, 257–275 (2003)
R.W. Heath, A.J. Paulraj, Linear dispersion codes for MIMO systems based on frame theory. IEEE Trans. Signal Process. 50, 2429–2441 (2002)
M.R. Abdollahpour, M.H. Faroughi, Generalized continuous frames in Hilbert spaces. Southeast Asian Bull. Math. 32, 1–19 (2008)
S.T. Ali, J.P. Antoine, J.P. Gazeau, Continuous frames in Hilbert spaces. Ann. Phys. 222, 1–37 (1993)
J.P. Gabardo, D. Han, Frames associated with measurable spaces. Adv. Comput. Math. 18, 127–147 (2003)
A. Rahimi, A. Najati, Y.N. Dehghan, Continuous frames in Hilbert spaces. Methods Funct. Anal. Topology 12, 170–182 (2006)
W.C. Sun, Generalized frames and g-Riesz bases. J. Math. Anal. Appl. 322, 437–452 (2006)
R. Balan, P.G. Casazza, D. Edidin, G. Kutyniok, Decompositions of frames and a new frame identity. Proc. SPIE. 5914, 379–388 (2005)
R. Balan, P.G. Casazza, D. Edidin, G. Kutyniok, A new identity for Parseval frames. Proc. Amer. Math. Soc. 135, 1007–1015 (2007)
P. Gävruta, On some identities and inequalities for frames in Hilbert spaces. J. Math. Anal. Appl. 321, 469–478 (2006)
X.G. Zhu, G.C. Wu, A note on some equalities for frames in Hilbert spaces. Appl. Math. Lett. 23, 788–790 (2010)
D.F. Li, W.C. Sun, Some equalities and inequalities for generalized frames. Chin. J. Contemp. Math. 29, 301–308 (2008)
J.Z. Li, Y.C. Zhu, Some equalities and inequalities for g-Bessel sequences in Hilbert spaces. Appl. Math. Lett. 25, 1601–1607 (2012)
X.C. Xiao, X.M. Zeng, Some equalities and inequalities of G-continuous frames. Sci. China Math. 53, 2621–2632 (2010)
O. Christensen, An Introduction to Frames and Riesz Bases (Birkhäuser, Boston, 2003)
K. Erwin, Introductory Functional Analysis with Applications (Wiley, New York, 1978)
Acknowledgements
The authors would like to express their gratitude to the referee for his (or her) valuable comments and suggestions.
This work is supported by National Natural Science Foundation of China (No.61471410).
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2022 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Li, D., Liu, L. (2022). Some Notes on the Inequalities for Parseval Generalized Continuous Frames. In: Cerejeiras, P., Reissig, M., Sabadini, I., Toft, J. (eds) Current Trends in Analysis, its Applications and Computation. Trends in Mathematics(). Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-87502-2_65
Download citation
DOI: https://doi.org/10.1007/978-3-030-87502-2_65
Published:
Publisher Name: Birkhäuser, Cham
Print ISBN: 978-3-030-87501-5
Online ISBN: 978-3-030-87502-2
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)