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A cardinal spline approach to wavelets

Authors: Charles K. Chui and Jian-zhong Wang
Journal: Proc. Amer. Math. Soc. 113 (1991), 785-793
MSC: Primary 41A15; Secondary 42C05
MathSciNet review: 1077784
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Abstract: While it is well known that the $ m$th order $ B$-spline $ {N_m}(x)$ with integer knots generates a multiresolution analysis, $ \cdots \subset {V_{ - 1}} \subset {V_0} \subset \cdots $, with the $ m$th order of approximation, we prove that $ \psi (x): = L_{2m}^{(m)}(2x - 1)$, where $ {L_{2m}}(x)$ denotes the $ (2m)$th order fundamental cardinal interpolatory spline, generates the orthogonal complementary wavelet spaces $ {W_k}$. Note that for $ m = 1$, when the $ B$-spline $ {N_1}(x)$ is the characteristic function of the unit interval $ [0,1)$, our basic wavelet $ {L'_2}(2x - 1)$ is simply the well-known Haar wavelet. In proving that $ {V_{k + 1}} = {V_k} \oplus {W_k}$, we give the exact formulation of $ {N_m}(2x - j), j \in \mathbb{Z}$, in terms of integer translates of $ {N_m}(x)$ and $ \psi (x)$. This allows us to derive a wavelet decomposition algorithm without relying on orthogonality nor construction of a dual basis.

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Keywords: Wavelets, cardinal splines, Euler-Frobenius polynomials, wavelet decompositions, algorithms
Article copyright: © Copyright 1991 American Mathematical Society

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