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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.

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
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