Determine the scaling functions and wavelet functions for the multi-wavelet basis in the Beylkin, Coifman and Rokhlin paper; consider multi-wavelets of order less than or equal to 2. Find the coefficients in the dilation and wavelet equations for these multi-wavelets. Program the cascade and pyramid algorithms for these multi-wavelets. Given f(x) = exp(x), determine the coefficients in the expansion of f on dyadic intervals of width 2^{-7} in terms of the multi-wavelet scaling functions. Then use the pyramid and cascade algorithms to determine the expansion of f in on coarser scales. Plot the multi-wavelet approximations on all scales.
Solution:
multiwaveletprob.pdf is the final report of this problem.
wavelet.pdf is a summary of wavelet lectures but NOT part of the problem. It may help clarify the concepts and notations I am using.
multiwavelet.nb is a Mathematica note book file that helps do certain calculations and draw figures.
multiwavelet.c is the C++ code that implements cascade and pyramid algorithms and decompose f(x) = exp(x) on different dyadic scales using multiwavelet of order M=3. The code initializes dilation coefficient d and wavelet coefficient w using the data in the final report "multiwaveletprob.pdf". It first decompose exp(x) on V_{-7} and output the coefficient a_{m,-7,k} on screen. It then does cascade algorithm to obtain a_{m,-j,k}and b_{m,-j,k} on coarse scales j=6,5,4,3,2,1,0. Then, it starts from a_{m,0,k} and b_{m,0,k} and use pyramid algorithm to get a_{m,-j,k} on finer scales j=1,2,3,4,5,6,7. (The correctness of these algorithms are demonstrated by the invariance of the coefficients.) The code finally output decomposition results on V_{-j} to file "fj.dat" of j=0,1,2,3,4,5,6,7 and results on V_{-j}+W_{-j}to file "dfj.dat" of j=0,1,2,3,4,5,6. Since V_{-j-1} = V_{-j}+W_{-j}, the two cases agree as shown in the report figure.