Chui, Charles K.; Wang, Jianzhong On compactly supported spline wavelets and a duality principle. (English) Zbl 0759.41008 Trans. Am. Math. Soc. 330, No. 2, 903-915 (1992). The authors consider a multiresolution analysis of \(L^ 2(\mathbb{R})\) generated by the \(m\)-th order \(B\)-spline \(N_ m(x)\). Then they describe the wavelet subspaces \(W_ k\) and construct a basic spline-wavelet with compact support that generates \(W_ k\). By Fourier transform techniques one can handle the corresponding pair of two-scale relations and a decomposition formula. Here the main idea is a duality principle which states that the pair of two-scale relations can be used as the decomposition formula, and vice versa. This approach yields a very desirable decomposition of every \(L^ 2\)-function into a direct sum of compactly supported spline wavelets. Reviewer: J.Prestin (Rostock) Cited in 4 ReviewsCited in 117 Documents MSC: 41A15 Spline approximation 42C05 Orthogonal functions and polynomials, general theory of nontrigonometric harmonic analysis 41A05 Interpolation in approximation theory 41A30 Approximation by other special function classes Keywords:multiresolution analysis; Fourier transform techniques; wavelets × Cite Format Result Cite Review PDF Full Text: DOI References: [1] Charles K. Chui, Multivariate splines, CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 54, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1988. With an appendix by Harvey Diamond. · Zbl 0687.41018 [2] Charles K. Chui and Jian-zhong Wang, A cardinal spline approach to wavelets, Proc. Amer. Math. Soc. 113 (1991), no. 3, 785 – 793. · Zbl 0755.41008 [3] 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 [4] Stephane G. Mallat, Multiresolution approximations and wavelet orthonormal bases of \?²(\?), Trans. Amer. Math. Soc. 315 (1989), no. 1, 69 – 87. · Zbl 0686.42018 [5] I. J. Schoenberg, Cardinal spline interpolation, Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1973. Conference Board of the Mathematical Sciences Regional Conference Series in Applied Mathematics, No. 12. · Zbl 0264.41003 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.