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Series Representation of Daubechies' Wavelets

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@Article{JCM-15-81,
author = {},
title = {Series Representation of Daubechies' Wavelets},
journal = {Journal of Computational Mathematics},
year = {1997},
volume = {15},
number = {1},
pages = {81--96},
abstract = { This paper gives a kind of series representation of the scaling functions $\phi_N$ and the associated wavelets $\psi _N$ constructed by Daubechies. Based on Poission summation formula, the functions $\phi_N(x+N-1), \phi_N(x+N), \cdots, \phi_N(x+2N-2) (0 \leq x\leq 1)$ are linearly represented by $\phi_N(x), \phi_N(x+1), \cdots, \phi_N(x+2N-2)$ and some polynomials of order less than $N$, and $\Phi _0(x):=(\phi_N(x), \phi_N(x+1), \cdots,\phi_N(x+N-2))^t$ is translated into a solution of a nonhomogeneous vector--valued functional equation$f(x)=A_d f(2x-d)+P _d(x),x\in [\frac{d}{2},\frac{d+1}{2}], d=0,1,$ where $A_0,A_1$ are $(N-1)\times (N-1)$--dimensional matrices, the components of$P_0(x), P_1(x)$ are polynomials of order less than $N$. By iteration,$\Phi_0(x)$ is eventualy represented as an $(N-1)$--dimensional vector series $\sum\limits_{k=0} ^\infty u _k(x)$ with vector norm $\|U _k(x)\|\leq C\belta ^k$,where $\belta =\belta_N <1$ and $\belta_N \searrow 0$ as $N \rightarrow \infty$. },
issn = {1991-7139},
doi = {https://doi.org/},
url = {http://global-sci.org/intro/article_detail/jcm/9191.html}
}

TY - JOUR
T1 - Series Representation of Daubechies' Wavelets
JO - Journal of Computational Mathematics
VL - 1
SP - 81
EP - 96
PY - 1997
DA - 1997/02
SN - 15
DO - http://doi.org/
UR - https://global-sci.org/intro/article_detail/jcm/9191.html
KW - Wavelet
KW - nonhomogeneous iterative scheme
AB - This paper gives a kind of series representation of the scaling functions $\phi_N$ and the associated wavelets $\psi _N$ constructed by Daubechies. Based on Poission summation formula, the functions $\phi_N(x+N-1), \phi_N(x+N), \cdots, \phi_N(x+2N-2) (0 \leq x\leq 1)$ are linearly represented by $\phi_N(x), \phi_N(x+1), \cdots, \phi_N(x+2N-2)$ and some polynomials of order less than $N$, and $\Phi _0(x):=(\phi_N(x), \phi_N(x+1), \cdots,\phi_N(x+N-2))^t$ is translated into a solution of a nonhomogeneous vector--valued functional equation$f(x)=A_d f(2x-d)+P _d(x),x\in [\frac{d}{2},\frac{d+1}{2}], d=0,1,$ where $A_0,A_1$ are $(N-1)\times (N-1)$--dimensional matrices, the components of$P_0(x), P_1(x)$ are polynomials of order less than $N$. By iteration,$\Phi_0(x)$ is eventualy represented as an $(N-1)$--dimensional vector series $\sum\limits_{k=0} ^\infty u _k(x)$ with vector norm $\|U _k(x)\|\leq C\belta ^k$,where $\belta =\belta_N <1$ and $\belta_N \searrow 0$ as $N \rightarrow \infty$.

X. G. Lu. (1970). Series Representation of Daubechies' Wavelets.

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*Journal of Computational Mathematics*.*15*(1). 81-96. doi: