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Stability of wavelet frames and Riesz bases, with respect to dilations and translations

Author: Jing Zhang
Journal: Proc. Amer. Math. Soc. 129 (2001), 1113-1121
MSC (2000): Primary 42C15; Secondary 41A30
Published electronically: December 7, 2000
MathSciNet review: 1814149
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Abstract: We consider the perturbation problem of wavelet frame (Riesz basis) $\{{\psi_{j,k,a_0,b_0}\}}=\{a_0^{nj/2}\psi(a_0^jx-kb_0)\}$ about dilation and translation parameters $a_0$ and $b_0$. For wavelet functions whose Fourier transforms have small supports, we give a method to determine whether the perturbation system $\{\psi_{j,k,a,b_0}\}$ is a frame (Riesz basis) and prove the stability about dilation parameter $a_0$ on Paley-Wiener space. For a great deal of wavelet functions, we give a definite answer to the stability about translation $b_0$. Moreover, if the Fourier transform $\hat{\psi}$ has small support, we can estimate the frame bounds about the perturbation of translation parameter $b_0$. Our methods can be used to handle nonhomogeneous frames (Riesz basis).

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  • [Ba] Balan, R., Stability Theorems for Fourier Frames and Wavelet Riesz Basis, The Journal of Fourier Analysis and Applications, 3(1997),499-504. MR 99f:42060
  • [B] Benedetto, J. J, Irregular sampling and frames, in ``Wavelets, a Tutorial in Theory and Applications" (Chui, C.K.Ed), 445-507, Academic Press, San Diego, 1992. MR 93c:42030
  • [BHW] Benedetto, J. J, Heil, C. and Walnut, D. F, Differentiation and the Balian-Low Theorem. J. Fourier Anal 1(1995),355-402. MR 96f:42002
  • [BW] Benedetto, J. J and Walnut, D. F, Gabor frames for $L^2$ and related spaces. in ``Wavelets:Mathematics and Applications" (J. J. Benedetto and M. Frazier, Eds), 97-162. CRC Press, Boca Raton, FL, 1994. MR 94i:42040
  • [C] Christensen, O., Frame perturbation, Proc. of Amer. Math. Soc. 123(1995), 1217-1220.
  • [CH] Christensen, O. and Heil, C. E., Perturbations of Banach Frames and atomic decompositions, Math. Nachr. 185(1997), 33-47. MR 98m:42061
  • [CS] Chui, C. K and Shi, X., Inequalities of Littlewood-Paley type for frames and wavelets, SIAM J. Math. Anal. 24(1993), 263-277. MR 94d:42039
  • [D1] Daubechies, I., ``Ten Lectures on Wavelets" SIAM, Philadelphia, 1992. MR 93e:42045
  • [D2] Daubechies, I., The wavelet transform, time-frequency localization and sigual analysis, IEEE Trans. Inform. Theory 36(1990), 961-1005. MR 91e:42038
  • [FG] Feichtinger, H. G. and Grächenig, Banach spaces related to integrable group representations and their atomic decompositions,1, J. Funct. Anal. 86(1989), 307-340. MR 91g:43011
  • [FZ] Favier, S. J. and Zalik, R. A, On the stability of frames and Riesz basis, Appl. Comput. Harm. Anal. 2(1995), 160-173. MR 96e:42030
  • [HW] Heil, C. E. and Walnut, D. F., Continuous and discrete wavelet transforms, SIAM Rev. 31(1989), 628-666. MR 91c:42032
  • [L] Long Ruilin, ``Hidimensional Wavelet Analysis", High Educational Press, Beijing (in Chinese)1995.
  • [Y] Young, R. M., ``An Introduction to Nonharmonic Fourier Series", Academic Press, New York, 1980. MR 81m:42027
  • [Z1] Zhang Jing, On the stability of wavelet and Gabor frames (Riesz bases), J of Fourier Anal. 1(1999), 105-125. MR 2000a:42055
  • [Z2] Zhang Jing, The existence of frames due to nonhomogeneous Calder ${\acute{o}}$n reproducing formula, preprint.

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Additional Information

Jing Zhang
Affiliation: Institute of Mathematics, Academia Sinica, Beijing, People’s Republic of China 100080
Address at time of publication: Department of Mathematics, Washington University, Campus Box 1146, One Brookings Drive, St. Louis, Missouri 63130-4899

Keywords: Frames, stability, wavelets
Received by editor(s): July 13, 1998
Received by editor(s) in revised form: July 1, 1999
Published electronically: December 7, 2000
Dedicated: Dedicated to the memory of Professor Long Ruilin
Communicated by: Christopher D. Sogge
Article copyright: © Copyright 2000 American Mathematical Society

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