Volume 15, Issue 1
Series Representation of Daubechies' Wavelets
DOI:

J. Comp. Math., 15 (1997), pp. 81-96

Published online: 1997-02

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• 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 $N \rightarrow \infty$.

• Keywords

Wavelet nonhomogeneous iterative scheme

@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 $N \rightarrow \infty$.