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Truncations of generalized shift-invariant systems

Part of: Nontrigonometric harmonic analysis

Published online by Cambridge University Press:  29 November 2023

Ole Christensen Open the ORCID record for Ole Christensen [Opens in a new window] ,
Pablo Garcia Alvarez  and
Rae Young Kim Open the ORCID record for Rae Young Kim [Opens in a new window]
Ole Christensen
Affiliation:
Department of Applied Mathematics and Computer Science, Technical University of Denmark, Building 303, 2800 Lyngby, Denmark e-mail: [email protected]
Pablo Garcia Alvarez
Affiliation:
Instituto de Matemática Aplicada San Luis, Universidad Nacional de San Luis and CONICET, Av. Ejército de Los Andes 950, D5700 HHW San Luis, Argentina e-mail: [email protected]
Rae Young Kim*
Affiliation:
Department of Mathematics, Yeungnam University, 280 Daehak-Ro, Gyeongsan, Gyeongbuk 38541, Republic of Korea
*
e-mail: [email protected]
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Abstract

We provide conditions under which a generalized shift-invariant (GSI) system can be approximated by a GSI system for which the generators have compact support in the Fourier domain. The approximation quality will be measured in terms of the Bessel bound (upper frame bound) for the difference between the two GSI systems. In particular, this leads to easily verifiable conditions for a perturbation of a GSI system to preserve the frame property.

Keywords

Generalized shift-invariant systemapproximation of generalized shift-invariant systemframes

MSC classification

Primary: 42C15: General harmonic expansions, frames 42C40: Wavelets and other special systems
Type
Article
Information
Canadian Mathematical Bulletin , Volume 67 , Issue 2 , June 2024 , pp. 504 - 515
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Copyright
© The Author(s), 2023. Published by Cambridge University Press on behalf of The Canadian Mathematical Society

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Footnotes

This research was supported by the Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (2016R1D1A1B02009954); and the 2018 Yeungnam University Research Grant.

References

Benavente, A., Christensen, O., Hasannasab, M., Kim, H. O., Kim, R. Y., and Kovac, F. D., Approximately dual pairs of wavelet frames . J. Math. Anal. Appl. 507(2022), Article no. 125841, 11 pp.CrossRefGoogle Scholar
Christensen, O., An introduction to frames and Riesz bases. 2nd expanded ed., Birkhäuser, Boston, 2016.Google Scholar
Christensen, O. and Hasannasab, M., Frame representations via suborbits of bounded operators . Stud. Math. 263(2022), 1–18.CrossRefGoogle Scholar
Christensen, O. and Rahimi, A., Frame properties of wave packet systems in ${L}^2({R}^d)$ . Adv. Comput. Math. 29(2008), 101–111.CrossRefGoogle Scholar
Hernandez, E., Labate, D., and Weiss, G., A unified characterization of reproducing systems generated by a finite family II . J. Geom. Anal. 12(2002), 615–662.CrossRefGoogle Scholar
Labate, D., Weiss, G., and Wilson, E., An approach to the study of wave packet systems. Wavelets, frames and operator theory . Contemp. Math. 345(2004), 215–235.CrossRefGoogle Scholar
Ron, A. and Shen, Z., Generalized shift-invariant systems . Constr. Approx. 22(2005), 1–45.CrossRefGoogle Scholar