×

zbMATH — the first resource for mathematics

A unified characterization of reproducing systems generated by a finite family. (English) Zbl 1029.42026
Let \(C\) be a nonsingular \(n\times n\) matrix with real entries. Given a collection of functions \(\{g_p\}_{p\in P}\) in \(L^2(\mathbb{R}^n)\), tight frames of the form \(\{g_p(\cdot -Ck)\}_{k\in \mathbb{Z}^n, p\in P}\) are characterized. Furthermore, equivalent conditions for two systems of this type to be dual frames are given. The results are applied to wavelet systems and Gabor systems.

MSC:
42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems
42C15 General harmonic expansions, frames
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Bonami, A., Soria, F., and Weiss, G. Band-limited wavelets,J. Geom. Anal.,3(6), 544–578, (1993). · Zbl 0811.42012
[2] Bownik, M. A characterizations of affine dual frames in L2(\(\mathbb{R}\)n),Appl. Comput. Harmon. Anal.,8(2), 203–221, (2000). · Zbl 0961.42018
[3] Bownik, M. The structure of shift-invariant subspaces spaces in L2(\(\mathbb{R}\)n),J. Funct. Anal,177(2), 282–309, (2000). · Zbl 0986.46018
[4] Bownik, M. On characterizations of multiwavelets in L2(\(\mathbb{R}\)n),Proc. Am. Math. Soc.,129, 3265–3274, (2001). · Zbl 0979.42021
[5] Calogero, A. A Characterization of wavelets on general lattices,J. Geom. Anal.,10(4), 597–622, (2000). · Zbl 1057.42025
[6] Casazza, P., Christensen, O., and Janssen, A.J.E.M. Classifying Tight Weyl-Heisenberg Frames, The functional and harmonic analysis of wavelets and frames, (San Antonio, TX, 1999),Contemp. Math.,247, Am. Math. Soc., Providence, RI, 131–148, (1999). · Zbl 0960.42006
[7] Chui, C., Czaja, W., Maggioni, M., and Weiss, G. Characterization of general tight wavelets frames with matriz dilations and tightness preserving oversampling,J. Fourier Anal Appl., to appear, (2001). · Zbl 1005.42020
[8] Chui, C., Shi, X., and Stöckler, J. Affine frames, quasi-affine frames and their duals,Adv. Comp. Math.,8, 1–17, (1998). · Zbl 0892.42019
[9] Czaja, W. Characterizations of Gabor systems via the Fourier transform,Collect. Math.,51(2), 205–224, (2000). · Zbl 0966.42021
[10] Daubechies, I.Ten Lectures on Wavelets, CBMS-NSF Regional Conference Series in Applied mathematics, Vol. 61, SIAM, Philadelphia, (1992).
[11] Daubechies, I., Jaffard, S., and Journès, J. A simple Wilson ortonormal basis with exponential decay,SIAM J. Math. Anal,22, 554–572, (1991). · Zbl 0754.46016
[12] Daubechies, I., Landau, H., and Landau, Z. Gabor time-frequency lattices and the Wexler-Raz identity,J. Fourier Anal. Appl.,1, 437–478, (1995). · Zbl 0888.47018
[13] DeBoor C., DeVore, R.A., and Ron, A. The structure of finetely generated shift-invariant spaces in L2(\(\mathbb{R}\)n),J. Funct. Anal.,119, 37–78, (1994). · Zbl 0806.46030
[14] Duffin, R.J. and Schaeffer, A.C. A class of nonharmonic Fourier series,Trans. Am. Math. Soc.,72, 341–366, (1952). · Zbl 0049.32401
[15] Frazier, M., Garrigós, G., Wang, K., and Weiss, G. A characterization of functions that generate wavelets and related expansions,J. Fourier Anal. Appl.,3, 883–906, (1997). · Zbl 0896.42022
[16] Gressman, P., Labate, D., Weiss, G., and Wilson, E.N. Affine, quasi-affine and co-affine and wavelets, inBeyond Wavelets, Stöckler, J. and Welland, G.V., Eds., Academic Press, (2001) · Zbl 1087.65125
[17] Gröchenig, K. and Haas, A. Self-similar lattice tilings,J. Fourier Anal. Appl.,2, 131–170, (1994). · Zbl 0978.28500
[18] Heil, C. and Walnut, D. Continuous and discrete wavelet transforms,SIAM Rev.,31(4), 628–666, (1989). · Zbl 0683.42031
[19] Helson, H.Lectures on Invariant Subspaces, Academic Press, New York, (1994). · Zbl 0119.11303
[20] Hernández, E. and Weiss, G.A First Course on Wavelets, CRC Press, Boca Raton, FL, (1996). · Zbl 0885.42018
[21] Janssen, A.J.E.M. Signal analystic proofs of two basic results on lattice expansions,Appl. Comput. Harm. Anal.,1, 350–354, (1994). · Zbl 0834.42019
[22] Janssen, A.J.E.M. Duality and biorthogonality for Weyl-Heisenberg frames,J. Fourier Anal. Appl.,4, 403–436, (1995). · Zbl 0887.42028
[23] Janssen, A.J.E.M. The duality condition for Weyl-Heisenberg frames, inGabor Analysis and Algorithms: Theory and Applications, Feichtinger, H.G., and Ströhmer, T., Eds., Birkhäuser, Boston, 33–84, (1998).
[24] Rieffel, M.A. Von Neumann algebras associated with pair of lattices in Lie groups,Math. Ann.,257, 403–418, (1981). · Zbl 0486.22004
[25] Ron, A. and Shen, Z. Frames and stable bases for shift-invariant subspaces of L2(\(\mathbb{R}\)d),Can. J. Math.,47, 1051–1094, (1995). · Zbl 0838.42016
[26] Ron, A. and Shen, Z. Weyl-Heisenberg frames and Riesz bases in L2(\(\mathbb{R}\)d),Duke Math. J.,89, 237–282, (1997). · Zbl 0892.42017
[27] Ron, A. and Shen, Z. Affine systems in L2(\(\mathbb{R}\)d): The analysis of the analysis operator*,J. Funct. Anal.,148, 408–447, (1997). · Zbl 0891.42018
[28] Rzeszotnik, Z. Calderón’s condition and wavelets,Collect. Math.,52(2), (2001). · Zbl 0989.42017
[29] Wexler, J. and Raz, S. Discrete Gabor expansions,Signal Processing,21, 207–221, (1990).
[30] Weiss, G. and Wilson, E.N. The mathematical theory of wavelets, inProceedings of the NATO-ASI meeting: Harmonic Analysis 2000-A Celebration, Kluwer, (2001).
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.