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On compactly supported spline wavelets and a duality principle. (English) Zbl 0759.41008
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.

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
Full Text: DOI
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