Gröchenig, Karlheinz; Ron, Amos Tight compactly supported wavelet frames of arbitrarily high smoothness. (English) Zbl 0911.42014 Proc. Am. Math. Soc. 126, No. 4, 1101-1107 (1998). Based on Ron and Shen’s new method for constructing tight wavelet frames [A. Ron and Z. Shen, J. Funct. Anal. 148, No. 2, 408-447 (1997; Zbl 0891.42018)], the authors show how to construct, for any dilation matrix, and any spatial dimension, tight wavelet frames generated by compactly supported functions with arbitrarily high smoothness. Reviewer: Richard A.Zalik (Auburn University) Cited in 31 Documents MSC: 42C40 Nontrigonometric harmonic analysis involving wavelets and other special systems 42C30 Completeness of sets of functions in nontrigonometric harmonic analysis Keywords:affine systems; frames; tight frames; multiresolution analysis; wavelets Citations:Zbl 0891.42018 × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Carl de Boor, Ronald A. DeVore, and Amos Ron, The structure of finitely generated shift-invariant spaces in \?\(_{2}\)(\?^{\?}), J. Funct. Anal. 119 (1994), no. 1, 37 – 78. · Zbl 0806.46030 · doi:10.1006/jfan.1994.1003 [2] Albert Cohen and Ingrid Daubechies, Nonseparable bidimensional wavelet bases, Rev. Mat. Iberoamericana 9 (1993), no. 1, 51 – 137. · Zbl 0792.42021 · doi:10.4171/RMI/133 [3] C.K. Chui, X.L. Shi and J. Stöckler, Affine frames, quasi-affine frames and their duals, CAT Report 372, Texas A&M University, College Station, TX, 77843, June 1996. · Zbl 0892.42019 [4] Stephan Dahlke, Wolfgang Dahmen, and Vera Latour, Smooth refinable functions and wavelets obtained by convolution products, Appl. Comput. Harmon. Anal. 2 (1995), no. 1, 68 – 84. · Zbl 0810.42020 · doi:10.1006/acha.1995.1006 [5] Ingrid Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988), no. 7, 909 – 996. · Zbl 0644.42026 · doi:10.1002/cpa.3160410705 [6] Ingrid Daubechies, Ten lectures on wavelets, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 61, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1992. · Zbl 0776.42018 [7] Karlheinz Gröchenig and Andrew Haas, Self-similar lattice tilings, J. Fourier Anal. Appl. 1 (1994), no. 2, 131 – 170. · Zbl 0978.28500 · doi:10.1007/s00041-001-4007-6 [8] K. Gröchenig and W. R. Madych, Multiresolution analysis, Haar bases, and self-similar tilings of \?\(^{n}\), IEEE Trans. Inform. Theory 38 (1992), no. 2, 556 – 568. · Zbl 0742.42012 · doi:10.1109/18.119723 [9] A. Ron and Z. Shen, Affine systems in \(L_{2}(\mathbb{R}^d)\), the analysis of the analysis operator, J. Functional Anal., to appear. Ftp site: anonymous@ftp.cs.wisc.edu/Approx file affine.ps [10] A. Ron and Z. Shen, Compactly supported tight affine spline frames in \(L_{2}(\mathbb{R}^d)\), Math. Comp., to appear. Ftp site: anonymous@ftp.cs.wisc.edu/Approx file tight.ps [11] Robert S. Strichartz, Wavelets and self-affine tilings, Constr. Approx. 9 (1993), no. 2-3, 327 – 346. · Zbl 0813.42021 · doi:10.1007/BF01198010 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.