# zbMATH — the first resource for mathematics

Compactly supported tight affine spline frames in $$L_{2}(\mathbb{R}^{d})$$. (English) Zbl 0892.42018
Summary: The theory of fiberization is applied to yield compactly supported tight affine frames (wavelets) in $$L_{2}(\mathbb{R}^{d})$$ from box splines. The wavelets obtained are smooth piecewise-polynomials on a simple mesh; furthermore, they exhibit a wealth of symmetries, and have a relatively small support. The number of “mother wavelets”, however, increases with the increase of the required smoothness.
Two bivariate constructions, of potential practical value, are highlighted. In both, the wavelets are derived from four-direction mesh box splines that are refinable with respect to the dilation matrix $$\left(\begin{smallmatrix} 1 & 1 \\ 1 & -1\end{smallmatrix}\right)$$.

##### MSC:
 42C15 General harmonic expansions, frames 41A15 Spline approximation 42C30 Completeness of sets of functions in nontrigonometric harmonic analysis 41A63 Multidimensional problems
Full Text:
##### References:
 [1] C. de Boor, K. Höllig, and S. Riemenschneider, Box splines, Applied Mathematical Sciences, vol. 98, Springer-Verlag, New York, 1993. · Zbl 0814.41012 [2] Albert Cohen and Ingrid Daubechies, Nonseparable bidimensional wavelet bases, Rev. Mat. Iberoamericana 9 (1993), no. 1, 51 – 137. · Zbl 0792.42021 [3] C.K. Chui and X. Shi, Inequalities on matrix-dilated Littlewood-Paley energy functions and oversampled affine operators, SIAM J. Math. Anal. 28 (1997), 213-232. CMP 97:06 · Zbl 0869.42011 [4] Charles K. Chui, Kurt Jetter, and Joachim Stöckler, Wavelets and frames on the four-directional mesh, Wavelets: theory, algorithms, and applications (Taormina, 1993) Wavelet Anal. Appl., vol. 5, Academic Press, San Diego, CA, 1994, pp. 213 – 230. · Zbl 0877.42015 [5] Ingrid Daubechies, The wavelet transform, time-frequency localization and signal analysis, IEEE Trans. Inform. Theory 36 (1990), no. 5, 961 – 1005. · Zbl 0738.94004 [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] Ingrid Daubechies, Orthonormal bases of compactly supported wavelets, Comm. Pure Appl. Math. 41 (1988), no. 7, 909 – 996. · Zbl 0644.42026 [8] Christopher E. Heil and David F. Walnut, Continuous and discrete wavelet transforms, SIAM Rev. 31 (1989), no. 4, 628 – 666. · Zbl 0683.42031 [9] Amos Ron and Zuowei Shen, Frames and stable bases for shift-invariant subspaces of \?$$_{2}$$(\?^{\?}), Canad. J. Math. 47 (1995), no. 5, 1051 – 1094. · Zbl 0838.42016 [10] A. Ron and Z. Shen, Affine systems in $$L_{2}(\mathbb R^d)$$: the analysis of the analysis operator, J. Func. Anal., to appear. Ftp site: anonymous@ftp.cs.wisc.edu
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.