No Access

PERTURBED FRAME SEQUENCES: CANONICAL DUAL SYSTEMS, APPROXIMATE RECONSTRUCTIONS AND APPLICATIONS

Departamento de Matemática, Facultad de Ciencias Exactas y Naturales, Universidad de Buenos Aires and IMAS-CONICET, Buenos Aires, Argentina

Search for more papers by this author

EWA MATUSIAK

Faculty of Mathematics, Universität Wien, NuHAG, Vienna, Austria

Search for more papers by this author

, and

VICTORIA PATERNOSTRO

Technische Universität Berlin, Institut für Mathematik, Berlin, Germany

Search for more papers by this author

https://doi.org/10.1142/S0219691314500192Cited by:3 (Source: Crossref)

Abstract

We consider perturbation of frames and frame sequences in a Hilbert space ℋ. It is known that small perturbations of a frame give rise to another frame. We show that the canonical dual of the perturbed sequence is a perturbation of the canonical dual of the original one and estimate the error in the approximation of functions belonging to the perturbed space. We then construct perturbations of irregular translates of a bandlimited function in L²(ℝ^d). We give conditions for the perturbed sequence to inherit the property of being Riesz or frame sequence. For this case we again calculate the error in the approximation of functions that belong to the perturbed space and compare it with our previous estimation error for general Hilbert spaces.

Keywords:

AMSC: 42C40, 42C15

References

A. Aldroubi and K. Gröchenig, SIAM Rev. 43(4), 585 (2001), DOI: 10.1137/S0036144501386986. Crossref, Web of Science, Google Scholar
P. Balazset al., Current Develop. Theory Appl. Wavelets 5(3), 165 (2011). Web of Science, Google Scholar
P. Balazs and S. Heineken, An operator theory approach to irregular frames of translates, preprint (2013) , arXiv: 1401.4955 . Google Scholar
J. Benedetto, A. Powell and O. Yilmaz, IEEE Trans. Inf. Theory 52(5), 1990 (2006), DOI: 10.1109/TIT.2006.872849. Crossref, Web of Science, Google Scholar
A. Beurling, Local harmonic analysis with some applications to differential operators, Proc. Ann. Science Conf., Belfer Graduate School of Science (1966) pp. 109–125. Google Scholar
S. Bishopet al., Linear Algebra Appl. 432(6), 1501 (2010), DOI: 10.1016/j.laa.2009.11.011. Crossref, Web of Science, Google Scholar
E. Candès and D. Donoho, Appl. Comput. Harmon. Anal. 19, 198 (2005). Crossref, Web of Science, Google Scholar
P. G. Casazza and O. Christensen, J. Fourier Anal. Appl. 3, 543 (1997). Crossref, Web of Science, Google Scholar
P. G. Casazza and O. Christensen, J. Approx. Theory 103(2), 338 (2000), DOI: 10.1006/jath.1999.3350. Crossref, Web of Science, Google Scholar
O. Christensen, Proc. Amer. Math. Soc. 123, 2199 (1995), DOI: 10.1090/S0002-9939-1995-1246520-X. Crossref, Web of Science, Google Scholar
O. Christensen , An Introduction to Frames and Riesz Bases , Applied and Numerical Harmonic Analysis ( Birkhäuser , 2003 ) . Crossref, Google Scholar
O. Christensen and C. Heil, Math. Nachr. 185, 33 (1997), DOI: 10.1002/mana.3211850104. Crossref, Web of Science, Google Scholar
O. Christensenet al., J. Geom. Anal. 15(2), 181 (2005), DOI: 10.1007/BF02922191. Crossref, Web of Science, Google Scholar
O. Christensen and R. S. Laugesen, Sampl. Theory Signal Image Process. 9, 77 (2011). Crossref, Web of Science, Google Scholar
O. Christensen, C. Lennard and C. Lewis, Rocky Mountain J. Math. 30(4), 1237 (2000), DOI: 10.1216/rmjm/1021477349. Crossref, Web of Science, Google Scholar
I. Daubechies, IEEE Trans. Inform. Theory 36(5), 961 (1990), DOI: 10.1109/18.57199. Crossref, Web of Science, Google Scholar
M. Dörfler and E. Matusiak, Nonstationary Gabor frames: existence and construction, preprint, to appear in Int. J. Wavelets Multiresolut. Int. Process . Google Scholar
R. Duffin and A. Schaeffer, Trans. Amer. Math. Soc. 72, 341 (1952), DOI: 10.1090/S0002-9947-1952-0047179-6. Crossref, Web of Science, Google Scholar
Y. C. Eldar and O. Christensen, EURASIP J. Appl. Signal Process. 7, (2006). Google Scholar
S. Favier and R. Zalik, Appl. Comput. Harmon. Anal. 2(2), 160 (1995), DOI: 10.1006/acha.1995.1012. Crossref, Web of Science, Google Scholar
H. G. Feichtinger, U. Molter and J. L. Romero, Int. J. Wavelets Multiresolut. Inf. Process. 6(2), 249 (2008), DOI: 10.1142/S0219691308002331. Link, Web of Science, Google Scholar
C. Heil, Y. Y. Koo and J. K. Lim, Acta Appl. Math. 107(3), 75 (2009), DOI: 10.1007/s10440-008-9410-4. Crossref, Web of Science, Google Scholar
T. Kato , Perturbation Theory for Linear Operators ( Springer , 1976 ) . Crossref, Google Scholar
Y. Y. Koo and J. K. Lim, Linear Algebra Appl. 420(3), 295 (2007), DOI: 10.1016/j.laa.2006.07.009. Crossref, Web of Science, Google Scholar
T. Strohmer and R. W. Heath, Appl. Comput. Harmon. Anal. 14(3), 257 (2003), DOI: 10.1016/S1063-5203(03)00023-X. Crossref, Web of Science, Google Scholar