Abstract
In this article we study finitely generated wavelet systems with arbitrary dilation sets. In 2002 Hernández et al. gave a characterization of when such a system forms a Parseval frame, assuming that a certain hypothesis known as the local integrability condition (LIC) holds. We show that, under some mild regularity assumption on the wavelets, the LIC is solely a density condition on the dilation sets. Using this new interpretation of the LIC, we further discuss when the characterization result holds.
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References
Casazza, P.G. and Kovaĉevié, I. Equal norm tight frames, with erasures.Adv Comput. Math. 18, 387–430, (2003).
Chan, R. H., Riemenschneider, S. D., Shen, I., and Shen, Z. Tight frame: An efficient way for high-resolution image reconstruction.Appl. Comput. Harmon. Anal. 17, 91–115, (2004).
Chui, C. K., Czaja, W., Maggioni, M., and Weiss, G. Characterization of general tight wavelet frames with matrix dilations and tightness preserving oversampling,J. Fourier Anal. Appl. 8(2), 173–200, (2002).
Chui, C. K., He, W., and Stöckler, J. Compactly supported tight and sibling frames with maximum vanishing moments,Appl. Comput. Harmon. Anal. 18, 224–262, (2002).
Daubechies, I.Ten Lectures on Wavelets, SIAM, Philadelphia, (1992).
Daubechies, I., Han, B., Ron, A., and Shen, Z. Framelets: MRA-based constructions of wavelet frames,Appl. Comput. Harmon. Anal. 14, 1–46, (2003).
Feichtinger, H. G. and Gröchenig, K. Banach spaces related to integrable group representations and their atomic decompositions, I.J. Funct. Anal. 86, 307–340, (1989).
Guo, K. and Labate, D. Some remarks on the unified characterization of reproducing systems,Collect. Math., to appear.
Heil, C. An introduction to weighted Wiener amalgams, inWavelets and their Applications (Chennai, January 2002), Krishna, M., Radha, R., and Thangavelu, S., Eds., Allied Publishers, New Delhi, 183–216, (2003).
Heil, C. and Kutyniok, G. Density of wavelet frames.J. Geom. Anal. 13(3), 479–493, (2003).
Heil, C. and kutyniok, G. The homogeneous approximation property for wavelet frames and Schauder bases, preprint, (2005).
Hernández, E., Labate, D., and Weiss, G. A unified characterization of reproducing systems generated by a finite family, II,J. Geom. Anal. 12(4), 615–662, (2002).
Kutyniok, G. Affine density, frame bounds, and the admissibility condition for wavelet frames,Constr. Approx., to appear.
Labate, D. A unified characterization of reproducing systems generated by a finite family,J. Geom. Anal.,12(3), 469–491, (2002).
Ron, A. and Shen, Z. Affine systems inL 2(ℝd): The analysis of the analysis operator.J. Funct. 148, 408–447, (1997).
Ron, A. and Shen, Z. Generalized shift-invariant systems,Constr. Approx. 22, 1–45, (2005).
Sun, W. and Zhou, X. Density and stability of wavelet frames,Appl. Comput. Harmon. Anal. 15, 117–133, (2003).
Sun, W. and Zhou, X. Density of irregular wavelet frames,Proc. Amer. Math. Soc. 132, 2377–2387, (2004).
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Communicated by Guido Weiss
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Kutyniok, G. The local integrability condition for wavelet frames. J Geom Anal 16, 155–166 (2006). https://doi.org/10.1007/BF02930990
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DOI: https://doi.org/10.1007/BF02930990