Abstract
The theory of superframes is widely used in multiplexing techniques. In this paper, we obtain a characterization of weak affine super bi-frames in reducing subspaces of \(L^{2}(\mathbb R,\,\mathbb C^{L})\). Our results are new even in \(L^{2}(\mathbb R,\,\mathbb C^L)\). Some examples are also provided.
Similar content being viewed by others
References
Atreas, N., Melas, A., Stavropoulos, T.: Affine dual frames and extension principles. Appl. Comput. Harmon. Anal. 36, 51–62 (2014)
Balan, R.: Density and redundancy of the noncoherent Weyl–Heisenberg superframes. Contemp. Math. 247, 29–41 (1999)
Bildea, S., Dutkay, D.E., Picioroaga, G.: MRA super-wavelets. N. Y. J. Math. 11, 1–19 (2005)
Bownik, M.: A characterization of affine dual frames in \(L^{2}(\mathbb{R}^n)\). Appl. Comput. Harmon. Anal. 8, 203–221 (2000)
Bownik, M., Rzeszotnik, Z.: Construction and reconstruction of tight framelets and wavelets via matrix mask functions. J. Funct. Anal. 256, 1065–1105 (2009)
Casazza, P.G., Christensen, O.: Weyl–Heisenberg frames for subspaces of \(L^{2}(\mathbb{R})\). Proc. Am. Math. Soc. 129, 145–154 (2001)
Casazza, P.G., Kutyniok, G.: Frames of subspaces. Contemp. Math. 345, 87–113 (2004)
Dai, X., Diao, Y., Gu, Q.: Subspaces with normalized tight frame wavelets in \(\mathbb{R}\). Proc. Am. Math. Soc. 130, 1661–1667 (2001)
Dai, X., Diao, Y., Gu, Q.: On super-wavelets. Oper. Theory Adv. Appl. 149, 153–165 (2004)
Dai, X., Diao, Y., Gu, Q., Han, D.: Frame wavelets in subspaces of \(L^{2}(\mathbb{R}^{d})\). Proc. Am. Math. Soc. 130, 3259–3267 (2002)
Dai, X., Diao, Y., Gu, Q., Han, D.: Wavelets with frame multiresolution analysis. J. Fourier Anal. Appl. 9, 39–48 (2003)
Dai, X., Diao, Y., Gu, Q., Han, D.: The existence of subspace wavelet sets. J. Comput. Appl. Math. 155, 83–90 (2003)
Daniel, A.L.: On the structure of Gabor and super Gabor spaces. Monatsh. Math. 161, 237–253 (2010)
Daubechies, I., Han, B.: Pairs of dual wavelet frames from any two refinable functions. Constr. Approx. 20, 325–352 (2004)
Daubechies, I., Han, B., Ron, A., Shen, Z.: Framelets: MRA-based constructions of wavelet frames. Appl. Comput. Harmon. Anal. 14, 1–46 (2003)
Dutkay, D.E., Jorgensen, P.: Oversampling generates super-wavelets. Proc. Am. Math. Soc. 135, 2219–2227 (2007)
Ehler, M.: On multivariate compactly supported bi-frames. J. Fourier Anal. Appl. 13, 511–532 (2007)
Ehler, M., Han, B.: Wavelet bi-frames with few generators from multivariate refinable functions. Appl. Comput. Harmon. Anal. 25, 407–414 (2008)
Führ, H.: Simultaneous estimates for vector-valued Gabor frames of Hermite functions. Adv. Comput. Math. 29, 357–373 (2008)
Gabardo, J.P., Han, D.: Subspace Weyl–Heisenberg frames. J. Fourier Anal. Appl. 7, 419–433 (2001)
Gabardo, J.P., Han, D.: Balian–Low phenomenon for subspace Gabor frames. J. Math. Phys. 45, 3362–3378 (2004)
Gabardo, J.P., Han, D.: The uniqueness of the dual of Weyl–Heisenberg subspace frames. Appl. Comput. Harmon. Anal. 17, 226–240 (2004)
Gabardo, J.P., Han, D., Li, Y.-Z.: Lattice tiling and density conditions for subspace Gabor frames. J. Funct. Anal. 265, 1170–1189 (2013)
Gröchenig, K., Lyubarskii, Y.: Gabor (super) frames with Hermite functions. Math. Ann. 345, 267–286 (2009)
Gu, Q., Han, D.: Super-wavelets and decomposable wavelet frames. J. Fourier Anal. Appl. 11, 683–696 (2005)
Gu, Q., Han, D.: Wavelet frames for (not necessarily reducing) affine subspaces. Appl. Comput. Harmon. Anal. 27, 47–54 (2009)
Gu, Q., Han, D.: Wavelet frames for (not necessarily reducing) affine subspaces II: the structure of affine subspaces. J. Funct. Anal. 260, 1615–1636 (2011)
Han, D., Larson, D.: Frames, bases and group representations. Mem. Am. Math. Soc. 147, 94 (2000)
Jia, H.-F., Li, Y.-Z.: Refinable function-based construction of weak (quasi-)affine bi-frames. J. Fourier Anal. Appl. 20, 1145–1170 (2014)
Jia, H.-F., Li, Y.-Z.: Weak (quasi-)affine bi-frames for reducing subspaces of \(L^2(\mathbb{R}^d)\). Sci. China Math. 58, 1005–1022 (2015)
Li, Z., Han, D.: Constructing super Gabor frames: the rational time-frequency lattice case. Sci. China Math. 53, 3179–3186 (2010)
Li, Y.-Z., Jia, H.-F.: Weak Gabor bi-frames on periodic subsets of the real line. Int. J. Wavelets Multiresolut. Inf. Process. 13, 1550046, 23 pp (2015)
Li, Y.-Z., Jia, H.-F.: Weak nonhomogeneous wavelet bi-frames for reducing subspaces of Sobolev spaces. Numer. Funct. Anal. Optim. 28, 181–204 (2017)
Li, Y.-Z., Zhang, Y.: Rational time-frequency vector-valued subspace Gabor frames and Balian–Low Theorem. Int. J. Wavelets Multiresolut. Inf. Process. 11, 1350013, 23 pp (2013)
Li, Y.-Z., Zhou, F.-Y.: Rational time-frequency super Gabor frames and their duals. J. Math. Anal. Appl. 403, 619–632 (2013)
Ron, A., Shen, Z.: Affine systems in \(L^{2}(\mathbb{R}^d)\): the analysis of the analysis operator. J. Funct. Anal. 148, 408–447 (1997)
Seip, K.: Regular sets of sampling and interpolation for weighted Bergman spaces. Proc. Am. Math. Soc. 117, 213–220 (1993)
Stavropoulos, T.: The geometry of extension principles. Houston J. Math. 38, 833–853 (2012)
Tian, Y., Li, Y.-Z.: Subspace dual super wavelet and Gabor frames. Sci. China Math. 60, 2429–2446 (2017)
Zhou, F.-Y., Li, Y.-Z.: Multivariate FMRAs and FMRA frame wavelets for reducing subspaces of \(L^{2}(\mathbb{R}^d)\). Kyoto J. Math. 50, 83–99 (2010)
Acknowledgements
This work was supported by National Natural Science Foundation of China (Grant No. 11271037). The authors would like to thank the referees for their comments which greatly improve the quality and readability of this article.
Author information
Authors and Affiliations
Corresponding author
Additional information
Supported by the National Natural Science Foundation of China (Grant No. 11271037).
Rights and permissions
About this article
Cite this article
Li, YZ., Tian, Y. Weak Affine Super Bi-frames for Reducing Subspaces of \(L^{2}(\mathbb R,\,\mathbb C^{L})\). Results Math 73, 96 (2018). https://doi.org/10.1007/s00025-018-0857-y
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s00025-018-0857-y