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Operators with Closed Range, Pseudo-Inverses, and Perturbation of Frames for a Subspace

Published online by Cambridge University Press:  20 November 2018

Ole Christensen
Ole Christensen*
Affiliation:
Department of Mathematics Technical University of Denmark Building 303 DK-2800 Lyngby Denmark, email: olechr@mat.dtu.dk
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Abstract

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Recent work of Ding and Huang shows that if we perturb a bounded operator (between Hilbert spaces) which has closed range, then the perturbed operator again has closed range. We extend this result by introducing a weaker perturbation condition, and our result is then used to prove a theorem about the stability of frames for a subspace.

Keywords

42C15
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 42 , Issue 1 , 01 March 1999 , pp. 37 - 45
Copyright
Copyright © Canadian Mathematical Society 1999

References

[B] Balan, R., Stability theorems for Fourier frames and wavelet Riesz bases. J. Fourier Anal. Appl. (5) 3 (1997), 499–504.Google Scholar
[BL] Benedetto, J. and Li, S., The theory of multiresolution frames and applications to filter design. Appl. Comput. Harmon. Anal., to appear 1998.Google Scholar
[CC1] Casazza, P. G. and Christensen, O., Perturbation of operators and applications to frame theory. J. Fourier Anal. Appl. (5) 3 (1997), 543–557.Google Scholar
[CC2] Casazza, P. G. and Christensen, O., Frames containing a Riesz basis and preservation of this property under perturbations. SIAM J. Math. Anal. (1) 29 (1998), 266–278.Google Scholar
[CCK] Casazza, P. G., O. Christensen and N. Kalton, Frames of translates. Preprint.Google Scholar
[C1] Christensen, O., Frame perturbations. Proc. Amer.Math. Soc. 123 (1995), 1217–1220.Google Scholar
[C2] Christensen, O., A Paley-Wiener theorem for frames. Proc. Amer. Math. Soc. 123 (1995), 2199–2202.Google Scholar
[C3] Christensen, O., Frames and pseudo-inverses. J. Math. Anal. Appl. 195 (1995), 401–414.Google Scholar
[C4] Christensen, O., Frames containing a Riesz basis and approximation of the frame coefficients using finite dimensional methods. J. Math. Anal. Appl. 199 (1996), 256–270.Google Scholar
[CDH] Christensen, O., Deng, B. and Heil, C., Density of Gabor frames. Appl. Comput. Harmon. Anal., to appear.Google Scholar
[CFL] Christensen, O., deFlitch, C. and Lennard, C., Perturbation of frames for a subspace of a Hilbert space. Submitted.Google Scholar
[CH] Christensen, O. and Heil, C., Perturbation of Banach frames and atomic decomposition. Math. Nachr. 185 (1997), 33–47.Google Scholar
[DH1] Ding, J. and Huang, L. J., On the perturbation of the least squares solutions in Hilbert spaces. Linear Algebra. Appl. 212–213 (1994), 487–500.Google Scholar
[DH2] Ding, J. and Huang, L. J., Perturbation of generalized inverses of linear operators in Hilbert spaces. J. Math. Anal. Appl. 198 (1996), 505–516.Google Scholar
[DS] Duffin, R. J. and Schaeffer, A. C., A class of nonharmonic Fourier series. Trans. Amer.Math. Soc. 72 (1952), 341–366.Google Scholar
[FZ] Favier, S. J. and Zalik, R. A., On the stability of frames and Riesz bases. Appl. Comput. Harmon. Anal. 2 (1995), 160–173.Google Scholar
[G] Goldberg, S., Unbounded linear operators. McGraw-Hill, New York, 1966.Google Scholar
[GZ] Govil, N. K. and Zalik, R. A., Perturbation of the Haar wavelet. Proc. Amer. Math. Soc. 125 (1997), 3363–3370.Google Scholar
[H] Heil, C., Wiener Amalgam spaces in generalized harmonic analysis and wavelet theory. Thesis, Univ. of Maryland, 1990.Google Scholar
[Ho] Holub, J., Pre-frame operators, Besselian frames and near-Riesz bases in Hilbert spaces. Proc. Amer. Math. Soc. 122 (1994), 779–785.Google Scholar
[HW] Heil, C. and Walnut, D., Continuous and discrete wavelet transforms. SIAM Rev. 31 (1989), 628–666.Google Scholar
[K] Kato, T., Perturbation theory for linear operators. Springer, New York, 1976.Google Scholar
[Y] Young, R., An introduction to nonharmonic Fourier series. Academic Press, New York, 1980.Google Scholar