Zhu, Yuan; Gao, Wenjun; Li, Dengfeng Characterization of generators for multiresolution analyses with composite dilations. (English) Zbl 1211.42037 Abstr. Appl. Anal. 2011, Article ID 850850, 13 p. (2011). Summary: This paper introduces multiresolution analyses with composite dilations (AB-MRAs) and addresses frame multiresolution analyses with composite dilations in the setting of reducing subspaces of \(L^2 (\mathbb R^n)\) (AB-RMRAs). We prove that an AB-MRA can induce an AB-RMRA on a given reducing subspace \(L^2(S)^{\vee}\). For a general expansive matrix, we obtain the characterizations for a scaling function to generate an AB-RMRA, and the main theorems generalize the classical results. Finally, some examples are provided to illustrate the general theory. MSC: 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems × Cite Format Result Cite Review PDF Full Text: DOI EuDML References: [1] E. Hernández and G. Weiss, A First Course on Wavelets, Studies in Advanced Mathematics, CRC Press, Boca Raton, Fla, USA, 1996. · Zbl 1194.65124 · doi:10.1016/0167-2789(96)00013-9 [2] D. F. Walnut, An Introduction to Wavelet Analysis, Applied and Numerical Harmonic Analysis, Birkhäuser, Boston, Mass, USA, 2004. · Zbl 0989.42014 [3] Q.-F. Lian and Y.-Z. Li, “Reducing subspace frame multiresolution analysis and frame wavelets,” Communications on Pure and Applied Analysis, vol. 6, no. 3, pp. 741-756, 2007. · Zbl 1141.42024 · doi:10.3934/cpaa.2007.6.741 [4] H. Y. Zhang and X. L. Shi, “Characterization of generators for multiresolution analysis,” Acta Mathematica Sinica. Chinese Series, vol. 51, no. 5, pp. 1035-1040, 2008. · Zbl 1174.42006 [5] K. Guo, D. Labate, W.-Q. Lim, G. Weiss, and E. Wilson, “Wavelets with composite dilations and their MRA properties,” Applied and Computational Harmonic Analysis, vol. 20, no. 2, pp. 202-236, 2006. · Zbl 1086.42026 · doi:10.1016/j.acha.2005.07.002 [6] K. Guo, D. Labate, W.-Q. Lim, G. Weiss, and E. Wilson, “Wavelets with composite dilations,” Electronic Research Announcements of the American Mathematical Society, vol. 10, pp. 78-87, 2004. · Zbl 1066.42023 · doi:10.1090/S1079-6762-04-00132-5 [7] K. Guo, D. Labate, W.-Q. Lim, G. Weiss, and E. Wilson, “The theory of wavelets with composite dilations,” in Harmonic Analysis and Applications, pp. 231-250, Birkhäuser, Boston, Mass, USA, 2006. · Zbl 1129.42433 · doi:10.1007/0-8176-4504-7_11 [8] O. Christensen, An Introduction to Frames and Riesz Bases, Applied and Numerical Harmonic Analysis, Birkhäuser, Boston, Mass, USA, 2002. [9] D. F. Li and M. Z. Xue, Bases and Frames on Banach Space, Science Press, Beijing, China, 2007. [10] X. Dai, Y. Diao, Q. Gu, and D. Han, “The existence of subspace wavelet sets,” Journal of Computational and Applied Mathematics, vol. 155, no. 1, pp. 83-90, 2003. · Zbl 1016.42020 · doi:10.1016/S0377-0427(02)00893-2 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. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.