Optimal wavelet expansion via sampled-data control theory

Wavelet theory provides a new type of function expansion and and has found many applications in signal processing. The discrete wavelet transform of a signal x(t) in L²(R) is usually computed by the so-called pyramid algorithm. It however requires a proper initialization, i.e., expansion coefficients with respect to the basis of one of the desirable approximation subspaces. An interesting question is how we can obtain such coefficients when only sampled values of x(t) are available. The paper provides a design method for a digital filter that optimally gives such coefficients assuming certain a priori knowledge on the frequency characteristic of the target functions. We then extend the result to the case of non-orthogonal wavelets. Examples show the effectiveness of the proposed method