On affine frames with transcendental dilations
HTML articles powered by AMS MathViewer
- by R. S. Laugesen PDF
- Proc. Amer. Math. Soc. 135 (2007), 211-216 Request permission
Abstract:
We answer a question of O. Christensen about affine systems in $L^2(\mathbb {R})$. Specifically, we show that if the dilation factor $a>1$ is transcendental, then cancellations cannot occur between different scales, in the conditions for the affine system to form a frame. Such cancellations are known to occur when $a$ is an integer.References
- R. Balan, P. Casazza and D. Edidin. On signal reconstruction without phase. Appl. Comput. Harmon. Anal. 20 (2006), 345–356.
- John J. Benedetto, Noise reduction in terms of the theory of frames, Signal and image representation in combined spaces, Wavelet Anal. Appl., vol. 7, Academic Press, San Diego, CA, 1998, pp. 259–284. MR 1614977, DOI 10.1016/S1874-608X(98)80010-1
- John J. Benedetto and Matthew Fickus, Finite normalized tight frames, Adv. Comput. Math. 18 (2003), no. 2-4, 357–385. Frames. MR 1968126, DOI 10.1023/A:1021323312367
- Marcin Bownik, Quasi-affine systems and the Calderón condition, Harmonic analysis at Mount Holyoke (South Hadley, MA, 2001) Contemp. Math., vol. 320, Amer. Math. Soc., Providence, RI, 2003, pp. 29–43. MR 1979930, DOI 10.1090/conm/320/05597
- Peter G. Casazza, The art of frame theory, Taiwanese J. Math. 4 (2000), no. 2, 129–201. MR 1757401, DOI 10.11650/twjm/1500407227
- Peter G. Casazza, Custom building finite frames, Wavelets, frames and operator theory, Contemp. Math., vol. 345, Amer. Math. Soc., Providence, RI, 2004, pp. 61–86. MR 2066822, DOI 10.1090/conm/345/06241
- Peter G. Casazza and Ole Christensen, Weyl-Heisenberg frames for subspaces of $L^2(\mathbf R)$, Proc. Amer. Math. Soc. 129 (2001), no. 1, 145–154. MR 1784021, DOI 10.1090/S0002-9939-00-05731-2
- Ole Christensen, Frames, Riesz bases, and discrete Gabor/wavelet expansions, Bull. Amer. Math. Soc. (N.S.) 38 (2001), no. 3, 273–291. MR 1824891, DOI 10.1090/S0273-0979-01-00903-X
- Charles K. Chui, Wojciech Czaja, Mauro Maggioni, and Guido Weiss, Characterization of general tight wavelet frames with matrix dilations and tightness preserving oversampling, J. Fourier Anal. Appl. 8 (2002), no. 2, 173–200. MR 1891728, DOI 10.1007/s00041-002-0007-4
- Charles K. Chui and Xian Liang Shi, Bessel sequences and affine frames, Appl. Comput. Harmon. Anal. 1 (1993), no. 1, 29–49. MR 1256525, DOI 10.1006/acha.1993.1003
- Ingrid Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory 36 (1990), no. 5, 961–1005. MR 1066587, DOI 10.1109/18.57199
- R. J. Duffin and A. C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952), 341–366. MR 47179, DOI 10.1090/S0002-9947-1952-0047179-6
- Karlheinz Gröchenig, Foundations of time-frequency analysis, Applied and Numerical Harmonic Analysis, Birkhäuser Boston, Inc., Boston, MA, 2001. MR 1843717, DOI 10.1007/978-1-4612-0003-1
- G. H. Hardy and E. M. Wright, An introduction to the theory of numbers, Oxford, at the Clarendon Press, 1954. 3rd ed. MR 0067125
- Eugenio Hernández, Demetrio Labate, and Guido Weiss, A unified characterization of reproducing systems generated by a finite family. II, J. Geom. Anal. 12 (2002), no. 4, 615–662. MR 1916862, DOI 10.1007/BF02930656
- Eugenio Hernández and Guido Weiss, A first course on wavelets, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1996. With a foreword by Yves Meyer. MR 1408902, DOI 10.1201/9781420049985
- Richard S. Laugesen, Completeness of orthonormal wavelet systems for arbitrary real dilations, Appl. Comput. Harmon. Anal. 11 (2001), no. 3, 455–473. MR 1866351, DOI 10.1006/acha.2001.0365
- Richard S. Laugesen, Translational averaging for completeness, characterization and oversampling of wavelets, Collect. Math. 53 (2002), no. 3, 211–249. MR 1940326
- Amos Ron and Zuowei Shen, Affine systems in $L_2(\mathbf R^d)$: the analysis of the analysis operator, J. Funct. Anal. 148 (1997), no. 2, 408–447. MR 1469348, DOI 10.1006/jfan.1996.3079
Additional Information
- R. S. Laugesen
- Affiliation: Department of Mathematics, University of Illinois, Urbana, Illinois 61801
- MR Author ID: 337007
- Email: Laugesen@uiuc.edu
- Received by editor(s): June 30, 2005
- Received by editor(s) in revised form: July 31, 2005
- Published electronically: June 29, 2006
- Additional Notes: This work was completed during a Visiting Erskine Fellowship at the University of Canterbury, New Zealand, and also with support from National Science Foundation award DMS–0140481.
- Communicated by: Joseph A. Ball
- © Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 135 (2007), 211-216
- MSC (2000): Primary 42C40; Secondary 42C15
- DOI: https://doi.org/10.1090/S0002-9939-06-08456-5
- MathSciNet review: 2280189