Abstract
In this paper we propose an algorithm for the calculation of the exact values of compactly-supported Daubechies wavelet functions. The algorithm is iterative, performing a single convolution operation at each step. It requires solving, at the first step only, a linear system of a relatively small size. The novelty of the algorithm is that once the values at dyadic points at a certain level j are calculated they do not need to be updated at the next step. We find that this algorithm is superior to the well-known cascade algorithm proposed by Ingrid Daubechies. This algorithm can serve well in wavelet based methods for the numerical solutions of differential equations. The algorithm is tested on Daubechies scaling functions as well as Daubechies coiflets. Comparison with the values obtained using the cascade algorithm is made. We found that the cascade algorithm results converge to ours
Highlights
The widespread interest in wavelets and their applications started in the 1980s after the breakthrough made by Daubechies [1,2] in constructing the first orthogonal compactly-supported wavelets with arbitrary regularity
The results clearly show that the cascade algorithm results converge to ours as j tends to infinity.We remark that one has to iterate the cascade algorithm for larger value of j to obtain accurate results at a lower k level dyadics
We have presented an efficient algorithm for the computation of the exact values of refinable functions, in particular Daubechies’ scaling and wavelet functions
Summary
The widespread interest in wavelets and their applications started in the 1980s after the breakthrough made by Daubechies [1,2] in constructing the first orthogonal compactly-supported wavelets with arbitrary regularity. [2] Daubechies described an algorithm known as the cascade algorithm for computing approximate values of the compactly-supported scaling and wavelet functions with arbitrary high precision. We propose an algorithm to calculate the exact values of Daubechies scaling and wavelet functions.
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