Oversampling and preservation of tightness in affine frames
Author: Raquel G. Catalán
Journal: Proc. Amer. Math. Soc. 130 (2002), 1031-1034
MSC (2000): Primary 42C40
Published electronically: October 12, 2001
MathSciNet review: 1873776
Full-text PDF Free Access
Abstract: The problem of how an oversampling of translations affects the bounds of an affine frame has been proposed by Chui and Shi. In particular, they proved that tightness is preserved if the oversampling factor is coprime with the dilation factor. In this paper we study, in the dyadic dilation case, oversampling of translation by factors which do not satisfy the above condition, and prove that tightness is preserved only in the case of affine frames generated by wavelets having frequency support with very particular properties.
-  Charles K. Chui and Xian Liang Shi, Bessel sequences and affine frames, Appl. Comput. Harmon. Anal. 1 (1993), no. 1, 29–49. MR 1256525, https://doi.org/10.1006/acha.1993.1003
-  Charles K. Chui and Xian Liang Shi, 𝑛× oversampling preserves any tight affine frame for odd 𝑛, Proc. Amer. Math. Soc. 121 (1994), no. 2, 511–517. MR 1182699, https://doi.org/10.1090/S0002-9939-1994-1182699-5
-  Ingrid Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. MR 1162107
-  Ingrid Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory 36 (1990), no. 5, 961–1005. MR 1066587, https://doi.org/10.1109/18.57199
-  Gustaf Gripenberg, A necessary and sufficient condition for the existence of a father wavelet, Studia Math. 114 (1995), no. 3, 207–226. MR 1338828
-  Christopher E. Heil and David F. Walnut, Continuous and discrete wavelet transforms, SIAM Rev. 31 (1989), no. 4, 628–666. MR 1025485, https://doi.org/10.1137/1031129
-  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
-  Yves Meyer, Ondelettes et opérateurs. II, Actualités Mathématiques. [Current Mathematical Topics], Hermann, Paris, 1990 (French). Opérateurs de Calderón-Zygmund. [Calderón-Zygmund operators]. MR 1085488
-  David F. Walnut, Continuity properties of the Gabor frame operator, J. Math. Anal. Appl. 165 (1992), no. 2, 479–504. MR 1155734, https://doi.org/10.1016/0022-247X(92)90053-G
-  -, Weil-Heisenberg wavelet expansion: existence and stability in weighted spaces, Ph. D. Thesis. University of Maryland (1989).
-  X.Wang, The study of wavelets from the properties of their Fourier Transforms, Ph.D.Thesis, Washington University in St. Louis (1995).
-  Robert M. Young, An introduction to nonharmonic Fourier series, Pure and Applied Mathematics, vol. 93, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. MR 591684
- C.K.Chui-X.Shi, Bessel sequences and Affine frames, Applied and Computational Harmonic Analysis, n. 1, pp. 29-49 (1993). MR 95b:42028
- C.K.Chui-X.Shi, oversampling preserves any tight affine frame for odd , Proceedings of the AMS, Vol. 121, n.2, pp. 511-517. (1994). MR 94h:42052
- I.Daubechies, Ten Lectures on Wavelets, CBMS, (1992). MR 93e:42045
- -, The wavelet transform, time-frequency localization, and signal analysis, IEEE Trans. Information Theory, Vol. 36, pp. 961-1005 (1990). MR 91e:42038
- G.Gripenberg, A necessary and sufficient condition for the existence of a father wavelet, Studia Math. 114(3), pp. 207-226 (1995). MR 96d:42049
- C.Heil, D.Walnut, Continous and discrete wavelet transform, SIAM Review, 31, pp. 628-666 (1989). MR 91c:42032
- E.Hernandez-G.Weiss, A first course on wavelets, CRC Press (1996). MR 97i:42015
- Y.Meyer, Ondelettes et operateurs,II, Hermann (1990). MR 93i:42003
- D.Walnut, Continuity properties of the Gabor frame operator, J. Math.Analysis. Appl. 165, pp. 479-504 (1992). MR 93f:42059
- -, Weil-Heisenberg wavelet expansion: existence and stability in weighted spaces, Ph. D. Thesis. University of Maryland (1989).
- X.Wang, The study of wavelets from the properties of their Fourier Transforms, Ph.D.Thesis, Washington University in St. Louis (1995).
- R. Young, An Introduction to Nonharmonic Fourier Series, Academic Press (1980). MR 81m:42027
Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 42C40
Retrieve articles in all journals with MSC (2000): 42C40
Raquel G. Catalán
Affiliation: Departamento de Matemática e Informática, Universidad Pública de Navarra, 31006, Pamplona, Spain
Keywords: Wavelets, frames, tight frames
Received by editor(s): September 3, 1999
Received by editor(s) in revised form: September 29, 2000
Published electronically: October 12, 2001
Additional Notes: This work was partially supported by the Spanish DGES PB97-1013, and originated during a stay at the Politecnico di Torino with the European TMR network on “Applications of the wavelet element method to boundary value problems".
Communicated by: David R. Larson
Article copyright: © Copyright 2001 American Mathematical Society