Abstract
The notion of quasi-affine frame was recently introduced by Ron and Shen [9] in order to achieve shift-invariance of the discrete wavelet transform. In this paper, we establish a duality-preservation theorem for quasi-affine frames. Furthermore, the preservation of frame bounds when changing an affine frame to a quasi-affine frame is shown to hold without the decay assumptions in [9]. Our consideration leads naturally to the study of certain sesquilinear operators which are defined by two affine systems. The translation-invariance of such operators is characterized in terms of certain intrinsic properties of the two affine systems.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
C.S. Burrus and R.A. Gopinath, Wavelet transforms and filter banks, in: Wavelets: A Tutorial in Theory and Applications, ed. C.K. Chui (Academic Press, Boston, 1992) pp. 603–654.
C.K. Chui, J.C. Goswami and A.K. Chan, Fast integral wavelet transform on a dense set of the time-scale domain, Numer. Math. 70 (1995) 283–302.
C.K. Chui and X. Shi, Bessel sequences and affine frames, Applied Computational Harmonic Analysis 1 (1993) 29–49.
C.K. Chui and X. Shi, Inequalities on matrix-dilated Littlewood–Paley energy functions and oversampled affine operators, SIAM J. Math. Anal. 28 (1997) 213–232.
X. Dai and D.R. Larson, Wandering vectors for unitary systems and orthogonal wavelets, to appear in Memoirs of the Amer. Math. Soc.
I. Daubechies, Ten Lectures on Wavelets, CBMS-NSF Reg. Conf. Series in Appl. Math. 61 (SIAM, Philadelphia, PA, 1992).
R.J. Duffin and A.C. Schaeffer, A class of nonharmonic Fourier series, Trans. Amer. Math. Soc. 72 (1952) 341–366.
E. Hernandez, X. Wang and G. Weiss, Smoothing minimally supported (MSF) wavelets I, to appear in J. Fourier Analysis and Applications.
A. Ron and Z. Shen, Affine systems in L 2(Rd): the analysis of the analysis operator, to appear in J. Functional Analysis.
M.J. Shensa, The discrete wavelet transform: wedding the à trous and Mallat algorithms, IEEE Trans. Signal Processing 40 (1992) 2464–2482.
J. Stöckler, A Laurent operator technique for multivariate frames and wavelet bases, in: Advanced Topics in Multivariate Approximation, eds. F. Fontanella, K. Jetter and P.-J. Laurent (World Scientific Publ., Singapore, 1996) pp. 339–354.
K. Yosida, Functional Analysis (Springer, Berlin, 6th ed., 1980).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Chui, C.K., Shi, X. & Stöckler, J. Affine frames, quasi-affine frames, and their duals. Advances in Computational Mathematics 8, 1–17 (1998). https://doi.org/10.1023/A:1018975725857
Issue Date:
DOI: https://doi.org/10.1023/A:1018975725857